Who calls fouls in ultimate frisbee?

Answer 1

The players.

More specifically, the 11th Edition rules for Ultimate state:

XVI. Violations and Fouls

H. Fouls (II.E): It is the responsibility of all players to avoid contact in every way possible .

1. A foul can be called only by the fouled player and must be announced by loudly calling foul immediately after it occurs.

Answer 2

Players call their own fouls so they can occur at anytime. When a foul does occur play is stopped, people stay right where they were, person throwing the disc gets is back, and play resumes as if nothing happened.