I work at Sonic & I sprained the inside of my ankle. Should I continue working with roller blades if I have a splint or should I stay off skates?

Probably best not to. There are many different degrees of ankle sprains. But if you are unable to walk on your toes( heels off the ground) without pain , a sensation of weakness you are not ready to put on the skates and or walk around without a brace on.
Ask mgr 4 alternate. There might be other tasks you can do while you heal. Ask the manager. I imagine Sonic would not like to be responsible for your doing further damage and it sounds like it is a risk. Also, I don't mean to be cavalier w/ your job - but it's not a critical first responder thing. If you don't heal well, your next 60-80 yrs quality of life could be at stake. I'd play it safe.