Being from the Midwest, I'm not fluent in all the snow in the mountains. I had initially planned a trip for Wyoming to Montana; but was encouraged to re-think that because of the weather.
If snow is a significant risk, I would likely change plans to drive from Salt Lake City to Phoenix instead, at that time. Would that be a safer plan?
There is no reason to rethink your initial road trip idea at the beginning of October, the northern Rockies usually enjoy nice Indian Summer weather well into October.
There is always a chance of a cold front coming through, perhaps even with snow, but usually the roads are clear shortly after the storm passes and at worst you might be delayed for a day.
While you may find some seasonal services shutting down in October, places like Yellowstone National Park itself stay open to vehicles until the end of October. And fall is a good time to see more wildlife in the parks.