Bits and pieces, here and there. Pretty much all live events are cancelled, postponed, or in a limbo state at the moment. The Chinese LOL League is still going; they're just playing online. Several CS:GO leagues are online-based, so they're still running. Some Dota2 qualifiers are supposed to be going on, though considering they're for LAN events whose fates are in question, I'm not sure how competitive they'll be. Fighting game tourneys will probably only cancel if they literally aren't allowed to run, since that's how the FGC is.
So yeah, even e-sports is pretty dire right now.