Finding enough apprentices to not only maintain roughly the same size, but to ensure the Order continues growing in size seems to be something generally brushed over within Sagas.
ME has a total population of roughly 68,000,000 people scattered over 4,066,000 square miles in 1220.
There are roughly 6,800 people with the Gift scattered about ME at the standard start date of 1220. Of those, roughly 1,360 will be in the age range or younger to be taken as a Hermetic Apprentice. While that might seem like plenty, we are talking about roughly one in every 50,000 people and one in roughly every 3,000 square miles.
Things get even rougher for most of the Order when you consider the population spread. France accounts for nearly a third of the total population, so its two Tribunals have an easier time. However there are several Tribunals in which the total Order membership in residence there is larger than the total Gifted individuals that occur due to their population.
While the Order is the largest organization of Gifted individuals in ME (accounting for almost 20% alive), even with their non-Gifted support personnel they are a small drop in the bucket of the total population. Additionally many members of the Order rarely if ever interact with mundanes or even travel. So how does an organization that must maintain roughly 20% of a very small and difficult to identify group actually find enough of that group to maintain itself.
How Can We Fix It (Without Major Changes)
So what are some ways we can fix the difficulty of all the Order who wishes an apprentice to gain one, without creating changes that would cause some drastic change in Hermetic culture?
To achieve this we need some method of testing thousands of youths spread out over thousands of miles, while not requiring some change in how Magi normally act. So no setting up Tribunal appointed jobs as "apprentice hunters" for Magi.
The easiest way I could find would be to task part of the mundane support in the Order who already travels all over with the job. The Redcaps seem to be the obvious choice here, though some Covenants who would rather local(ish) Gifted children will task a few of their Grogs with the duty.
So now that we have a rough idea of who will be conducting the search, we need to design a rough idea of how they conduct it. Magic, by enchanted items, is the most likely method. Without it they will almost never find those with the Gentle Gift.
The InVi Base 10 (Detect the Gift) enchanted into items is the easiest way. It will have to be more powerful than Individual and Touch (Group and Voice/Sight) or as Sense (Vision, Hearing, or Smell). The effect will generally be powerful enough to cause Warping, so Sense magic can be damaging for the user unless it is crafted specifically for them.
A Voice Range, Group Target version that only allows the user to detect who has the Gift in range is Level 30 plus Uses/Day modifier. Bumping up the Range to Sight would raise it to 35 plus Uses/Day modifier. Items like this (with possible levels of Size to improve the search speed) would be the most common since they do not have to be custom enchanted for their user.
If we are trying to avoid major changes, the Redcaps will only take note of all the Gifted children they find rather than trying to collect them all up.
In general play this should not have any noticeable effect. Magi who are looking for an Apprentice can still do the search on their own if desired. Or they can ask the Redcaps for assistance, paying some small nominal fee (a few pawns of Vis perhaps).
Some of the effects that will show up are that some portion of Redcaps will carry enchanted items for identifying the Gift. Creating these items will be a talk the Redcaps most likely hire out, with any Vis changed for the service used to create new items.
So a small portion of Redcaps have a task, a type of enchanted item that Redcaps might possess, and a new concept for Redcaps (Gifted Hunter).