What's available for a demon-hunter that doesn't have the Gift, after character creation? That is, besides a custom/homebrew initiation script.
Being ordained as an exorcist and obtaining a relic (and lots of holy water) would seem to be the best strategy.
There's a big difference between "mundane" and "not Magic." There's three other Realms to get demon-fighting power from, two of which are pretty easy to access. And it's not like the Infernal Realm is united enough to not help you.
Sense Holiness and Unholiness is the go-to power for spotting the Infernal.
True Faith give Magic Resistance and allows you to learn Holy Methods and Powers.
Guardian Angel can add to Soak and Magic Resistance.
Second Sight lets you see invisible/immaterial stuff.
Infernal Lore lets you figure out a demon's weaknesses.
To think a bit more about "After Character Creation" aspect:
- True Faith - sure, you could go on a pilgrimage to gain that.
- Sense Holiness/Unholiness - I'm guessing an Initiation into a Holy Order would be a requirement, here.
- Relic - a minor one would likely be the easiest one to get. Those fingerbones of St. FillInTheBlank and shards of the True Cross are all over the place. A Major one would be really nice, but...yeah. Probably a bit difficult to just walk off with.
- Infernal Lore - I'd argue that most books on Divine Lore also contain implicit Infernal Lore - but that's an issue of books teaching more than one topic. Alternately, the first two levels can probably be gotten fairly easily just with standard XP drops. (5xp + 10xp - would take 3 sessions, assuming you only spend 5 xp each time. That's a rule, isn't it?)
- Second Sight - I think it would have to be Divine-aspected to work against demons, right? Otherwise, an Initiation would also probably be required here.
- Guardian Angel - probably kind of difficult to receive, in game mechanics (again - aside from Initiation.)
Either way, something that can be achieved by a story. As you say, there are relics all over the place: if you are willing to pay for one, or steal one, or possibly perform some Worthy Service in return for one, they should be accessible.
Learning the exorcism rite doesn't technically require initiations or anything. You just need to know what to say and do: you don't need to be clergy or even Christian. It's (marginally) more effective if you are ordained as an exorcist though.
Pilgrimage and Mysticism are both options in lieu of Initiation, too.
(Also, you doubled-up on a number.)
To gear up a demon hunter (post creation):
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item of quality
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Charged item from Rusticani (not necessary Ex Miscellanea but simple edge wizard)
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animal of virtue (bloodhound to track demon), or plant7tree of virtue for protection or to harm demon.
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Formulae (A&A p70) can assist a demon hunter - it will never match the ability of a magus, but it gives access to semi-magical items without need to contact hermetic magus:
- Astrological inceptions can help uncover evil at work (if you consider that stars are beyong the lunar sphere and close to the realm of God, it may overcome the demon's ability to remain hidden - if you opt for that, make sure that the difficulty of the inception is high enough to remain a challenge for players - this is really a YSMV)
- Alchemical reagent can make alternative weapon to holy water
- Ligatures to assist in demon hunting (just before a fight, or to boost investigation roll)
Finally, initiation by Learned magician can grant (painfully because of the major flaw involved in each initiation) some limited magical powers.
If a demon hunter wants magical powers of their own, the best solution is probably Ars Goetia, which also doesn't require an Initiation. (It does involve committing mortal sins, but surely a demon hunter will survive to spend their final years in penance and die under extreme unction, right? 
)
Pay a hermetic mage to make you a DEO wand and/or an amulet of warding vs. demons.
...of course, that might run afoul of the whole "don't taunt the demons" rule in the Hermetic code - but a sufficiently-motivated/vengeful magi could be using the PC as their catspaw to get around that particular issue. "Hey, I'm not the one poking the infernal with a stick. It's that stupid mundane over there. How was I to know he was planning on storming the Gates of Hell with that Rod of Demonic Smiting? I just sold it to him for recreational/self-defense/ah, who am I kidding/entertainment value!"
That being said, you can probably get a nice discount on the expiry. I mean, it's not like you're going to need it for more than a couple of years...
Get any number of supernatural folks to buff you. Holy magic is the most obvious, mainly because they've got access to the Ceremony skill, which is crazy-good at stacking with itself. So it's really not all that hard (from a game mechanic standpoint) to get, say, a year-long blessing (or longer - they have access to the "until you commit a mortal sin" duration) out of those people.
And if you're a demon hunter, the holy magi of your relative religion will probably want to bless you - at least in theory. So from a story perspective, it's really all about finding them.
Prayer (for anyone) or Intercession of the Saints (for a Christian) as outlined in RoP:D. While it works a lot better if you've got True Faith, technically speaking anyone can do it. It may take a bit of preparation to make it likely to pull off, though.
Probably your best bet is going on a Pilgrimage to gain a level of True Faith - just because it acts like the Gift for Holy magic. As such, you could pick up 3 major and a minor (I think) Holy Virtues by osmosis, once you got it.
Take communion on a regular basis. (Again, from RoP:D) - it acts like a temporary point of Faith until you commit a Mortal Sin.
Know your Church calendar - Infernal power waxes and wanes according to the Holy days. Also, if you can time a prayer to a Saint on their Saint's Day, you get a bonus.
(And in looking over the Invoking a Saint rules) - OK, seriously: get your Charm ability up, and specialize in "Saints". Assuming you do the following:
Com +1
Charm +5 (Saints) +1
Minor Relic in possession +1
Minor Monetary Donation +1
Confidence +3
Faith (from relic or communion) +3
That's a +15 right there, vs. a target number of 15. Unless you botch the roll, you can call on the power of a Saint once per month. And assuming your Saint in question has the power to expel demons (Both St. Peter and St. Martin, two of the examples in the book, do), you can get an automatic lvl 75 or lvl 50 DEO strike against your target.
Also note that Faith points in a Relic renew every morning, whereas Confidence points renew...whenever. So Faith points are probably a bit better to spend.
This really doesn't work from an RP perspective.
Literally, in fact. Noble's Parma, but I recall the book explicitly stating that a pilgrimage taken with the goal to receive True Faith will never succeed at that goal. Of course, that's an RP limitation, so you could do it while it's your goal but not your character's, but I'm not sure how I feel about randomly ignoring your character's personality to get a quick power boost.
On that note, I disagree with everyone here. As strong as it is, True Faith is bad for a person who will be spending a lot of time in proximity to demons. True Faith is a thing characters have for the sake of itself; it comes from the only Realm where "I want to be powerful" can't be your reason for having it, and that makes it unreliable. Do you really want your power to leave you the instant you sin when you're constantly in proximity of, speaking to, and fighting creatures whose only real unified specialty is making nearby people sin?
If you're willing to become your enemy to fight them, Infernal power can be very good for fighting the Infernal. Then you only have to worry about losing your power if you walk into a church and confess, which it's very near impossible to do unintentionally.
Alternatively, Faerie power is reportedly decently awesome and not hard to get. It doesn't contain any valid means of detecting demons, but I've known Hermetic magi who refuse to be shackled by that, so a dedicated enough Faerie wizard could try. Or even somebody who just has some Faerie gifts.
As others have also mentioned, holy initiation works and is a much more reliable (but far less powerful) conduit for Holy abilities than going and gaining True Faith. You might want to focus on getting Detect Holiness/Unholiness and Divine Second Sight, but there are other useful things to be Initiated into if you can handle it.
While those are valid points, they're also metagame issues. There are a LOT of things I want my character to do that, were I actually the character, I would have no interest in doing. As such, It's perfectly valid to say "I would like my character to become a demon hunter - he'll probably need True Faith for that, so I'd like a story arc whereby he's confronted by a powerful demon. Even if he's successful in fighting off the demon, it will shake his faith. Afterwards, he'll go on a pilgrimage to question his relationship to the Divine. Assuming it works, I'd like him to get a level of True Faith out of it." That's standard 'working with the Storyteller to set up a story arc'.
That being said - yes. If the CHARACTER decided "hm, I need True Faith to be a demon hunter. I know! I'll go on a pilgrimage to get more power from God!" Yes. That probably won't work. It would certainly be an interesting story of hubris and failure, (and I assume being attacked by demons along the way), but doesn't have anything to do with the PLAYER's desire to get True Faith for their character.
I agree with you Kevin.
Also, I'm not convinced that True Faith actually gest you much of significance, in play, for a demon hunter, that having a relic doesn't also get you?
Having a point of actual True Faith allows you to learn Holy Magic by osmosis, rather than having to be Initiated into it. (ie, you can learn 3 Major Holy virtues + 1 minor Holy virtue) - or at the least makes them SIGNIFICANTLY easier to learn.
Just because I'm aparantly dense, could you give me a page reference for this?
I'll look it up when I get home - I believe it's in the Holy Magic section in RoP:D, where it talks about how characters learn Holy Magic traditions.