If needed the antibiotic will be given orally. Usually twice a day with a syringe. It's not a pleasant procedure and is best done with two people. I hope you won't have to do that. I have done it myself but not for a UTI.
It seems that most chinchillas are okay with the meds but some are not and I've seen many cases where that happens. They end up loosing the chinchilla because it goes into GI stasis and by the time they start treating it it's too late. I think in most cases it is preventable because every time I ask how long their chinchilla had been off its food they were unable to determine that because they were not monitoring and suspected it had been for two or more days. Make sure to get detailed instructions from your vet on hand feeding and the proper supplies, if possible if you ever are prescribed antibiotics, especially baytril. Some veterinarians will prescribe hand feeding as part of their diet during the course of meds to prevent stasis, especially, if they are on a heavy dose.