It's reasonable to believe that a 1st-level Monk trained at a monastary. It's also reasonable to believe that you, as DM, have not detailed every single monastary in the entire world.
Of course it is.
The last game I ran was
Traveller. The setting was a subsector in Ley sector, along the trailing border of the Third Imperium. That subsector alone has four worlds with populations in the tens of billions.
If a player wants to describe the Imperial Naval vessel or Scout survey ship or sector merchant liner her character served on for twenty pages, she is more than welcome to do so - as long as she is doing so for her own use, not mine.
Again, in the case of the dwarf from the Northern Wastes example upthread, it's established fact in the setting that there are icy wastes in the north and that dwarves live there. I have no problem with a player detailing the tribe if he wishes, because it's pretty safe to assume that I haven't detailed every tribe in every part of the world, but don't expect me as referee to do anything with it in-game. Friendships, loyalties, rivals, social groups - the one's that matter are the one's we develop in actual play.
Unless we start play before 1st-level, how did the newly created monk find training in-game? Or is every monk in the campaign pigeon-holed into the "self-trained prodigy" archetype?
I think I've answered this already.
And even if you have created every monastary in the known word, why would you turn down a good idea from a player who wishes to belong to an order that espouses virtues different from your creations?
I don't turn it down. I encourage the player to pursue it as an in-character goal.
For the game I'm prepping to run, there are more than
thirty orders of knighthood which a character may join - and if the player doesn't see one he likes, he's welcome to create something new in-game. If the character wants to serve in the royal army, there are
sixteen regiments from which to choose - and if that doesn't work for him, he can raise a mercenary company if he wishes. There are
nine gentlemen's clubs (and more coming) to which a character may belong, and the option to found a new club is always there as well. More than thirty bishops' curias, scores of chapters, and dozens of confraternities exist for the religious - and if the character wants to demonstrate his piety in ways that none of those fulfill, then establishing a new sodality is perfectly feasible.
Especially if that player does not wish to focus in-game time on forming his own order.
If the player doesn't want to use any of the setting details, express or implied,
and the player doesn't want to be bothered with creating something that fits his vision in actual play, then his expectations are probably not a good fit for the style of game I'm prepared to run.
I do my level best to set clear expectations from the giddyup, to give the players enough information to decide if this is something they would enjoy playing or not. If not, no harm, no foul. Every game isn't a good fit for every player.
