I've never had tennis elbow in my string elbow, but I have had it in my bow elbow. I think a major cause of it on the bow side is hand shock. The cumulative effect of small shocks from shooting many, many shots eventually can result in inflamed tendons. When I got it, it took quite a while to get rid of. You don't want to completely lay off doing things, because that can result in atrophy. You don't want to force yourself to keep on shooting the way you were either, or mask the symptoms with too much ibuprofen. So it's kind of a delicate balancing act to work your way through it. In addition to cutting back on my shooting for way longer than I wanted to, and switching to a bow with less hand shock, I found some exercises that were useful, such as the Thera Band Flex Bar (see Amazon), and some other exercises I picked up by doing an internet search on tennis elbow.
I've been free of it for several years now, and can comfortably shoot up to a hundred shots every other day or so without any pain. The other way it's a balancing act is that while it probably is an overuse issue, once you build yourself up to where you can shoot a lot of arrows without pain, it sort of helps to keep shooting a lot of arrows. I think it helps not to make any major changes, like dramatically increasing bow weight or switching to a bow with more hand shock.