I camp while working in the sense that I'm a work camper, which isn't quite the same as what you're proposing. I take jobs specifically meant for RVers, where a RV site + utilities are included along with pay for working a seasonal job. I've been doing this for about 2 years now and greatly enjoyed the lifestyle. I set up these jobs in advance through phone interviews so by the time one ends, the next is lined up and ready to go. I'll space the jobs out enough so that I have time to travel and explore between them, like taking multiple "vacations" a year.
Wyoming winters will be tough. When I worked at an Amazon warehouse in Kansas for two of their peak seasons, by the end of the season in December it would be getting below freezing at night regular, and occasionally not get above freezing during the day. My trailer is not a 4 season, and I needed two electric heaters inside to keep the indoor temp 50 degrees warmer than it was outside. That was fine when it was 25 degrees outside, once it got into single digits which happened 3 or so times last year it was pretty miserable and I couldn't keep my plumbing from freezing up. I wrote a post about it on my blog and if you want to message me I'll share the link, I don't think it's allowed on posts like this.
For me, it was worth it for the great pay and I only had to endure those really cold temps for December, but I would think long and hard about doing it for a while winter in a climate where single digit and colder temps are normal. Whatever you decide I wish you the best of luck.