Tendonitis is caused by either a new or repetitive movement, or both. As others have mentioned rest, and anti-inflammatories such as ibuprofen can help. The best is ice, in my opinion, because there are really no negative side effects.
Every one of the NSAID drugs will have negative side effects over time, but immediately people notice how tough they can be on the stomach, as well as constipation, etc. I have had some cortisone injections in that spot that helped, but it's a temporary fix if you don't know why the tendonitis happened in the first place.
My surgeon, as well as my physical therapist told me it's often caused by a "twisting motion" which is why tennis players, golfers, bowlers, mechanics can have trouble repeatedly over the years.
For bowlers, specifically, it's about clearing the thumb earlier so that as we rotate the fingers around (to the right for righties) the thumb isn't still in the thumb hole causing extra torque, so to speak, on the forearm and elbow area.
It sounds simple, but once you get the inflamed area so calm down, you should practice delaying your rotation until the thumb has at least started to exit the thumb hole. It's a split-second movement and requires one to relax to do it correctly, but it avoids the problem AND your game will improve at the same time. Most bowlers start turning way too early, frankly.
This also assumes your ball is fitted properly with appropriate thumb and finger pitches that allow you to bowl without excessive grip pressure, etc. That's another topic.
It's not just about letting the tendonitis heal. It's better to prevent it from happening again. Good luck.
