Nope, they're pre-set. How it works is that there are three "slots" for each type, are there are a few Pokemon in each slot. The game selects a type, and one Pokemon from each slot, and that's what you get in your Friend Safari.
So there is an element of randomness in it, but no, you can't get each and every different kind of Pokemon available through the Friend Safari. Serebii has a list of Pokemon available in the Friend Safari, be sure to check that out.