Well, the online version might fail on heavy computations to be honest, so it might be best to have it installed locally if you're planning to do some serious stuff. But for the first acquaintance, the online SymPy should be fine.
You can also use SymPy on https://CoCalc.com, and it will handle heavier computations. (Disclaimer: I created cocalc.)
I'm watching your Harvard talk currently. Very cool. Is there any offline support? E.g. if I want to edit a cocalc document on a plane and merge with collaborators afterwards?