There are plenty of monkeys. There is no shortage of rhesus monkeys. They can breed all they need for experiments.
The problems with using monkeys is that they take longer to breed and they have smaller litters, and of course, people feel a bond with monkeys that they do not feel for mice and rats.
Monkeys also are more expensive to maintain. They eat a lot more than mice, they take up more space, and PETA and others make a big stink over experiments that are pretty cruel.
If I were a lab technician, I would prefer working with squeaky little mice than larger hooting monkeys that are known to fling their poop at those they dislike.