I looked at the listing and carefully at the photos. One thing jumped out at me as a potentially major concern.
Originally Tuckers only came with manual transmissions. At some point (in the 1970's I believe) they started to offer automatics. Initially they used the Chrysler Loadflite transmission and later added the Allison AT-545 as an additional option.
The Chrysler Loadflite is derived from their Torqueflite transmission and is a heavier duty version. It was offered with either a long tailstock housing or a short one. Tucker used the short versions. The Allison AT-545 is a four speed non-overdrive transmission for medium duty trucks. It was very popular for many, many years.
A couple of years ago I had a conversation with Jeff Godard at Tucker. He was then the parts and service manager (now he's in sales). Our conversation drifted to automatic transmissions. He thinks they're a good option but had nothing positive to say about the Torqueflite. A few years ago they had a customer who had a broken tailshaft housing. (It's basically an aluminum casting that gets machined in the production process.) Anyway, they couldn't find a replacement anywhere, at any price.
Jeff talked a bit about automatics in Sno-Cats and said in an automotive application their duty cycle consists partially of coasting and partially being under load. But when installed in a Sno-Cat it's basically being used full time under load. That's much harder on the transmission and means one needs a heavier duty transmission.
So, back to this cat...
If you look at the shift lever in the cab it's the traditional automotive P-R-N-D-2-1. An AT-545 would have R-N-D-3-2-1. (The AT-545 does not have a parking pawl in the transmission. You leave it in neutral and use the parking brake.)
[Some AT-545's came with an option called "double neutral" There it would be PB-R-N-D-D3-D1, but Tucker did not, to the best of my knowledge, use the double neutral setup. In the PB position the transmission itself is in neutral and either a mechanical cable or electric solenoid operates the parking brake.]
Some Tuckers come with front blades and some are used to pull drags in a grooming application. In both of those instances the transmission must work harder. This machine was owned by Arizona Public Service, an electric utility so I doubt it was used to pull a drag. You may be okay with the Loadflite transmission especially if you don't add a front blade or pull a drag, but at least you now know about the potential problem.