First, I am not a doctor. And I don't claim any advice or cure. Disclaimer.......You are experiencing a form of tendinitis. Since you don't ride year round, it's not the snowmobiling causing the problems. Yes it does irritate it. I'm going to guess that it is something you are doing everyday that is keeping it from healing properly.
Forget all the stupid neoprene wrist and arm bands. They are intended to minimize shock from movement of your arm. In reality all they do is restrict blood flow to the area.
Forget the regimen of anti-inflammatories he probably recommended. They dehydrate and tighten the area even more. Everything they told me to do didn't work and seemed to make it worse.
I went through this years ago which is why I'm telling you this. I was/am an avid weight lifter and stay in shape. They problem wasn't the weights. It was stemming from the odd/ awkward lifting I was doing at work.
If you are doing any lifting, you must lighten up the load. Or, stop all together.
The absolute best thing I found was to stretch the muscles around the area. This includes your forearm all the way to your hand. Also your bicep and all the way through your shoulders.
Tendinitis is an overuse syndrome. A couple thoughts on this. Your muscles are extremely tight and pulling on your tendons causing the pain. Or, you are using your muscles awkwardly transferring to much force to your tendons. And not the muscle. Hopefully this helps you at least find the cause.
In order to keep riding through this. It is very important to stretch and warm up the area first. It wouldn't hurt to meet with a physical therapist for some stretching techniques as well as some rotational and range of movement exercises to help heal the area.
The bad news is that it took at least 2 years to heal and 10 years later I can still get it to flare up if I push myself to hard.