Nowadays, chaplains almost always come as loaners from various dioceses. They hit the seminaries hard, speaking to young men and asking them to consider candidacy. And that itself can be a chore, since it requires permission from your bishop.

I can't speak to any one religious order providing a steady supply, but I would be highly doubtful of any doing so, given the current numbers problem in most orders.

I don't think there are any orders that would absolutely forbid it, outside of the fact that it just isn't what the order does persay. Which is the situation with almost every order :). I know a couple former Jesuits chaplains, however.

I also don't believe the service draws many vocations from the inside, though I know it happens (I've met exactly one guy, of the many I know in seminary who currently have a commission). One of the 'strings' involved is that you have to have three years as a priest outside of the military before you can go on full-time active duty.

If anyone's actually interested, the way to do it is to approach the vocation director in your diocese and explain that you feel called to serve in the military. If you state it all out front and seriously feel called to it, most bishops will keep open to the possibility of releasing you when the time comes.

However, there are no guarantees. That's why every Mass on base I've ever been to includes a petition for bishops to release priests to chaplaincy.