You absolutely can do it! It may well be that you surprise yourself.
I'm training to be a yoga teacher and hoping to specialise in non-traditional groups - so often yoga doesn't reach the people who it could benefit the most, because it seems like it's 'not for people like me' or simply because of cost.
If you do a google search for 'fat yoga' (that's the phrase chosen by the people leading it!), you'll find a ton of inspirational material and images to show people likely to be much bigger or less fit than you, just rocking a crane pose 
However - I'd also be honest and say that you ARE likely to encounter some different challenges than, for example, a skinny ex-dancer. Your injury as well as your body shape will make some poses harder work than for others in the class (but don't worry, everyone has something they find tricky - and what you might lose in the bends, you could gain in the balances). Your existing fitness will certainly help, but do be aware that it can make for really tight muscles in some people, especially if you've never done much flexibility-related exercise, so don't expect to bend like a pretzel in the first month (or 5)! The final challenge is sometimes just in showing up and feeling comfortable when classes often are full of the lululemon stereotype. I'd suggest working out a yoga outfit you really like so you can feel comfortable - whatever you wear for the gym is probably great, but if you do choose leggings make sure they have decent coverage, as some brands go see through during downward dog 
, and if you are at all self conscious about your stomach, a top that won't fall down to your ears when you bend forward for a sun salute, plenty of us have felt that awkwardness in the early days!
Lastly, for anyone with any health issues, I'd always suggest trying to find the right teacher. A good teacher will be able to tailor classes to suggest alternatives when the overall routine won't suit your shoulder injury or your physique (and not just for you, but for anyone who has a sore hand, or tight hamstrings etc), and will make you feel welcomed at the start of the class, supported throughout and renewed at the end. It may take trying out a few teachers and/or styles before you find the right one for you. It is worth it though, I spent many years thinking yoga wasn't for me, just because I hadn't found the right place. Hope you can find your place!