swimming is the main recommended one in the military for limited/light duty profiles or those that can't run for whatever reason, for what that's worth at least
you could probably do some light plyometrics or boxjumps or whatever too, depending on hard you push that's pretty decent cardio as well as core/vertical work as well
shit just stand under a hoops backboard and quickly jump up and touch it repetitively as fast as you can. Kinda similar effect but not as crazy as Burpees since you said shoulder issues or whatnot
edit: i'm retarded, nevermind you probably can't reach up like that haha. try to hit your head on the indoor ceiling, same ish