Interesting. There's a list of archers and lancers at Agincourt online, but I don't think the complete lists of men are online.
Estimates of Fauconberg's DoB vary, but it looks like 1405 must be too late. The usual system then was that a gentleman with spare cash would contract to provide so many men for so many days, at his own expense, for which the king would pay a fixed price later. So the Captains selected themselves. Whether they went in person isn't always clear, but it seems unlikely they could be under age.
Regular musters were taken by Treasury staff to make sure the contracted numbers were delivered, hence these lists.
PS scratch that, as these musters aren't from Henry V's time, they're later.