Ouch man, I feel your pain. I’ve given myself both elbow and shoulder tendonitis multiple times by working out. However I am pretty sure the treatment would be the same as for RSI tendonitis, specifically:
1 - Most important - stop doing what you were doing to cause the injury. Not permanently, just for now. You don’t want to aggravate it. In this case, maybe see how it feels to switch to a pick on bass for a while, or see if it’s a good time to learn slap, if those feel better? In any case, ease off a bit until you can talk to a pro.
2 - Talk to a pro about it ASAP. Trainers or physical therapists would be perfect and could tell you if you need to see a doctor. They could also tell you alternate exercises to help.
3 - Don’t stop exercising, but work around it. Sounds like you have this one covered 
4 - If there is someone you can show your right hand plucking technique to they might be able to tell you what’s up. Josh and others have some good videos about it on youtube as well.
For a while I kept hurting myself working out because my workout involves heavy weight and lots of overhead pulling, kind of a recipe for shoulder and elbow injury. Plus I’m like forty-elevenish and not getting any younger. The above advice helped me.