Can Primates Learn to Code?

Obviously. Humans belong to order of primates.

No other species of primate has symbolic language as far as we can tell, so they probably can’t learn to code.

Depends on how simple your code is. I can imagine that a smart tool-using animal could be trained to use a mechanical computer or 'program' a simple sequence, but the prime motivator (that humans can supply) for most animals is food, so you would end up with a Rube-Goldberg food dispenser or something, not literal codemonkeys.