I had the same issues when I started a really good pullup program. I've found that warming up REALLY good has helped a lot. And I mean, I warm up almost to where I'm getting an arm workout, before hand. When you get up to speed, pullups can be a really good addition to your program.
I will keep in mind about the pullups, hopefully early next year I will be in a position to have access to a gym as well.
Unless you are at an elite level of athleticism or undergoing rehab with a qualified physical therapist, don't listen to anyone who gives you any exercise to specifically strengthen a tendon or ligament.
Work smartly for general strength. Use a RICER method if something hurts. Repeat as necessary. The strength of tendons and ligaments will take care of themselves.
I developed elbow tendonitis from improper framing so the same could be happening to you. Just watch how you use your forearms to defend in side control.