So...not sure if I simply haven't gotten far enough in the quests and stuff or if I'm just being oblivious to wherever it is but I can't find anything about getting more genetic scanners or if there's a place that does it or something. Any help would be appreciated.

After you get the freebie scanner/test kit, you pretty much don't get another one... unless you buy an unlimited kit with diamonds from the dia shop.

BUT! If you cannot buy diamonds, don't despair. Many people are willing to genetically test pets for free. If you look around the Clubs and Services forum (or whatever it's called), you'll probably find multiple threads for free testing.