If a nonGifted Redcap has half a dozen children, none of whom are gifted, can they apprentice all of them at the same time for introduction into the Order?
As far as I understand, each apprentice still has the right to one season of (solo) teaching each year.
I have the same interpretation. Plus the season(s) per year serving the house.
The RedCap could always have relatives take some of them as apprentice.
...or head on out to Transylvania, where customs differ.
... did the wife have a litter? Or possibly they are half-siblings from several women?
Regardless, I suspect that apprenticing your own children is not the best idea. Apprentice them out.
Isn't the best training to give to a redcap 'apprentice' the one on the road and in the business? Seeing other covenants, meeting and greeting their apprentices and magi, understanding the ways to approach them, tolerate their Gift and handle their requests, and avoiding trouble on the road: this is the practice that makes a redcap and gives her the Virtue Well Travelled.
So I don't see the redcap 'parens' put in more than two or three extra-seasons, beyond those on the road, to teach his ungifted pupil. But his work suffers, if he has a kindergarten in tow. And this limits his simultaneous 'apprentices' roughly like those of a Gifted magus.
Cheers
I think the apprentice does assist in the travelling duties, taking care of the bagages, meeting many covenants, etc. I also think he learns more from it than the "gifted apprentice who extracts vis", hence the virtue Well Travelled. However, I think the parens owes 1 season per year of proper teaching to the apprentice, as Tellus said. I don't believe the two could be combined. If they were, I think the RedCap would be slacking off on one or two of his duties.
Technically a brood in this instance.
A brood is more a general term for a group of offspring; a litter is from one birthing set. Neither term is very appropriate, so
... did she have sextuplets? 
FWIW, it wasn't all that long ago that families had lots of kids. My own grandmother was one of 14, although some did die at birth or within their first year, 9 survived to adulthood, and 6 of those made it to their 90s, including my grandmother.
Sure. My great-aunt had 13 children, including two in the same year, but not all at once. The OP wants to them all at once, which implies they are roughly the same age. I'm just joking about the sextuplets.
House Mercere has ... generous ... attitudes toward marriage, sex, and reproduction, so they might have all been born in the same two or three years. I thought it was a bit curious.
Back to point, p. 16 of Apprentices says that apprentices are due one season of training; it does not say they are due dedicated one-on-one training (although this is the only way to teach Arts). P. 38 of the same book says the master must do this personally (impacting my school model, I note). Perhaps there's a contradiction or clarification somewhere. It could be that a Redcap could train half-a-dozen, if they have the appropriate Teaching skill, but I still think training your own children would be a problem I would not want to take on.
Redcaps swear the Oath of Hermes (ArM5 p.13 box) and are considered magi by it. So they train apprentices and take responsibility for them.
They have no laboratories and cannot open or teach the Arts, so ArM5 p.106 Training your Apprentice and Apprentices p.32ff Hermetic Apprenticeship cannot apply literally. Most of p.38ff Legal Issues there, even the paragraph on the season of personal instruction by the parens, refers explicitly to Gifted children only.
Whether the Peripheral Code has special previsions for the training of un-Gifted redcaps is left to your saga - but their motivation would be different. An azzeccagarbugli Quaesitor overextending the Peripheral Code wrt redcap apprentices would have to overcome all of House Mercere: a very unlikely outcome.
Cheers
All with Strong Faerie Blood? Just wondering.
The real question is when would it come up? Not giving the season of training to a redcap is not depriving them of magical power. No Bonisuagus is going to snatch them to compensate for the abuse... the only ones who might care are other redcaps, and I suspect they settle the issue internally.
No sentence, no crime? 
I'm not sure on the proper term, clutch maybe? Either way, full disclosure, the character didn't actually have sextuplets, they had duodevigintuplets (?); eighteen children at once. Only 16 of them are potentially able to be apprenticed under her, as one is Gifted and another isn't sapient. I was using six as a hypothetical that was greater than four to at least fit the primary question whether the parent can get away with teaching more than one per season.
That is not stated in the Oath. It may be a code ruling in some tribunal. But is it in yours?? Who is watching you?? Who is going to care?? Who is stopping you???
No, it's not in the Oath, but it's clear that it's part of the Order-wide Peripheral Code.
How strictly is that enforced in various tribunals? And what of different traditions such as Transylvania? And who is paying attention to the...
wait, how many kids? Back this whole thing up. What happened? Juicy details please!
The character has 18, but only 16 are under consideration for being apprenticed because one isn't even human and another has the Gift. As for what happened, the character is a shapeshifting Redcap who took a fertility potion, then pretended to be a mundane and seduced a Bjornaer. While the guy lay in the forest, basking in the glow of a good time, the character snuck off and took the form of a grey partridge; which are known for having very large clutches of eggs. Dice were rolled, one hatched a regular bird, another hatched as a shapeshifter with the Gift, and the rest are 'mundane' skinchangers.