Good ole USA Doc!!!!!!!! I do believe that Tucker was one of the first ever made!!!!
Tucker was among the very first to attach tracks to a vehicle with skis on the front for steering. Pre-dating Tucker were snow-planes. Those were propeller driven vehicles that had 3 or 4 skis and were best suited to use on flat lands, frozen lakes, etc. I believe the earliest of those were the Russian Aerosans, dating back before 1920
(sometime in the "teens"). At least I can't find any record of any mechanically driven snow machines prior to the Russian Aerosan.
There were some "conversion" vehicles using Henry Ford's Model Ts with simple tracks added to the rear tires. Most commonly I see images of 3 axle conversions with the rear 2 axles holding traditional road tires, but with a simple wood & chain "track" arrangement that is utilized very similar to the track conversions you can find for modern skid-steers. The fronts of those model T conversions often had simple skis that were bolted onto the front axle and replaced the front wheels. We have photos of some of them here
(somewhere) on the forums.
The thing that seemed to make the Tuckers unusual was the early adoption of front and rear tracks. While most of the machines in the machines prior to the mid-1950's had skis on the front, Tucker was developing the 4 pontoon drive system.
In the 1940s and into the mid-1950s the propeller driven snow-planes were very popular. Fudge and a couple other companies dominated that category, most seem to be in the upper mid-west
(Minnesota, Dakotas, etc and also across the great planes of Canada). Other companies developed 2 track machines, and that is what happened with Snow Trac, but the Snow Trac was not commercially introduced until the very late 1950's and was not a commercially distributed unit until the early 1960's. Bombardier also developed a very successful 2 track series of machines, they were the "snow busses" and were the fastest commercial snowcats for a very long time. They were in production from the 1940s and into the 70's. They utilized 2 long rear tracks and 2 front skis. The fronts could also be equipped with traditional road wheels. They attained speeds exceeding 30mph, and could commonly seat 12 to 18 people
(depending on the model). The Bombardier B-12 through C-18 series were best suited for reasonably flat to mildly hilly lands and were very popular in Canada for use as school busses, mail delivery vehicles, etc.