The main thing to understand is that your are in charge of how you train. So if you would like to train light contact, or no contact at all, you should be able to. If your club does not respect that, they are not worthy: Martial Arts nowadays is not as it used to be in terms of need. We need it less for warfare and more for self-defence. As different people have different needs, if your Martial School does not have this awareness, they probably will not be very helpful to you.
I'll enumerate a couple of common injuries that are more or less associated with different styles.
Grappling - BJJ (of the grappling styles that's the one I have more experience with):
- Broken toes - this is due to being on the ground, getting swept.
- Cauliflower ears, just like rugby guys, due to friction of ears against Gi
- sore hip-flexors (sorry not too sure about the spelling) - this is because as a beginner of BJJ you will spend a lot of time on your back with the partner on your guard. This puts a lot of pressure on this area as you are tense and tend to squeeze your legs around the partners waist.
- Shoulders, some times due to locks applied to quickly before you have the chance to tap-out.
Judo - Knees for falling badly; broken fingers for getting it stuck inside your partner's Gi and blisters on your fingers due to friction when gripping.
Interestingly I have had less injuries with stand up fighting style like Muay Thai and Kickboxing, but the training is sorer because of the hits onto the face and body. Although people tend to think that BJJ is very safe, that's the art I got most injuries from.
Find training partners like-minded, people that appreciate the need to keep things safe and enjoy the training. Martial Arts' spectrum is so huge that you can adapt every training into your needs.
To sum up, I would take this approach:
Research the styles you like the most. Then, find out how the clubs in your place train. When speaking with the instructor, be clear to express your concerns and assess whether they have an open-mind towards your needs. Try to do your own researches on youtube and google about what your martial art is about. Then put an ad somewhere looking for partners or private instructors: as long as safety is your priority, the rest will flow. Remember, you don't need contact, strength or condition to simply train and enjoy martial arts as a hobby. In fact, something I always say: the slower you train (hence safer), the more control you gain (on a long run). The main think about Martial Arts is not about flying kicks or hitting hard; Martial Arts is all about being smart; so it's not about training hard, it is about training smart.