Hello, here are few species that live in desert climates. All of these are listed as LC - (Least Concern) by IUCN however, they may be protected within their range. Make sure that whatever species you decide on is legal to obtain.
Scaphiopus couchii - Couch's Spadefoot Toad
Hyla arenicolor - Canyon Treefrog
Pseudacris maculata - Boreal Chorus Frog