Moose Malone is [?was] an Australian blues-folk band.
There’s also a Scandi band called Wapiti.
If bands with ‘Bambi’ in their name count, there’s also:
The Bambi Molesters
Bambi Slam
Corey Hart, though not a band.
There are probably some variations on reindeer as a band name. Google points me to this one.
Hey, Hinds still counts! It’s just a fancier name for deer.
Buck Owens.
StG
If you’re willing to accept any surviving North American Even-Toed Ungulate rather than strictly deer, there’s Donna the Buffalo (clearly referring to bison)