My daughter had a full-arm cast due to a fracture and rode everything she wanted to, with the exception of two we decided were too bumpy (Goofy's Sky School and Matterhorn). She was only 43" tall, so she couldn't ride Indy or Califorina Screamin', but we would have let her ride CS with no problem. Indy is a bit jerky, so we would have thought long and hard about that one. Her doctor said it was fine to ride anything she wanted since the arm was protected in a cast, so I didn't worry too terribly much about anything other than rides that I personally had smacked an arm up against the ride vehicle while riding.
That said, if she had a cracked bone with just had a sling with no splint or anything else, I would have been much more cautious. There are quite a few rides with some pretty good side-to-side motion that I'd be worried about (Mater, Teacups, Indy). I wouldn't want to risk making the injury worse. That said, if it's more a muscular thing than a cracked bone, I would have let her ride anything she wanted.
There are also a couple of rides that might be a bit more difficult to get in and out of with only one arm. I can barely get out of the Astro Orbitor with two good arms!
Oh, CS does have and over-the-shoulder harness--that could be awkward, uncomfortable or even impossible with the sling.
Hope you have a great trip!