There are five (I think) mainstream rental outfits at the Durango airport (DRO). Budget's website indicates Wranglers for rent...but...
You should definitely call the actual Budget counter at DRO; don't rely on the website or the nationwide 1-800 number. We were out there with our Jeep a couple of years ago, and some friends with us had reserved a Wrangler from Budget using either the website or the 1-800 number. When we all showed up at the counter to pick up their rental Jeep, the Budget person said "we haven't had a Jeep here in months. Sorry..." They would have been happy to rent them a AWD Explorer or similar, telling them that it would in fact be an upgrade.
I would definitely talk to someone at the counter. You might try the others there also. They could possibly have a Wrangler or two even if their websites don't indicate it. The five agencies there are Budget, Hertz, Avis, National, and Dollar.
My family and I fly into DRO every winter to visit family and go skiing, so I just happen to have these numbers handy:
'03 red Rubi