Chinese repeating crossbow
I would like to raise what was already mentioned in passing by Alberto Yagos in a comment into a proper answer and build on it.
First a little bit about crossbows in general:
The crossbow was a scary weapon when it first appeared in Europe. It takes significantly less training to make a simple peasant adept at the weapon compared to a bow. Combine this with the fact that a single crossbow bolt can take out an armored noble at medium range. A feat that right before the crossbow entered was actually really difficult. Continuous improvements in armor-smithing together with proper layering with a gambaeson(impact reducing cloth armor worn under armor) an arrow would not penetrate unless it hit somewhere like the armpit. This and then the introduction of the first muskets would begin the decline full body armor in Europe.
I think you should consider the Chinese repeating crossbow (real name either Cho-ku-nu or Lián Nú). We know the first version was already in existence between 4th century BC so the technique for creating this weapon is definatley avaliable. It is effective enough that it was used for hundreds of years.
While it is neither as accurate or has as high impact as a normal crossbow an adept can fire large amounts of bolts quickly. It does however seem that the repeater was never used very much in favor for the 'normal' crossbow. Some of the issue does seem to stem from that the arrows for some models where not fletched. There is probably in theory a better design that can still utilize medieval craftsmanship if won where to combine the best aspects of the several types of repeater crossbows.
Combining repeating crossbows and normal crossbow units could perhaps give rather good ranged capacities to an army in short amount of time if the weapons are available but you lack trained soldiers. Repeaters for the output and even a small company of normal crossbow adepts becomes a real dangerous anti-armor addition to a medieval army.
But if just a large amount of projectiles in a short amount of time is what you want a professional archer during this era could nock and fire an arrow with incredible speed. Actually Legolas firing speed is about on par with this. It is e.g said it was an entry test for the renowned Saracen archers to fire 4 arrows before the first hit the ground something that has been proven to actually be possible.