The disposal is backing up but not the other sink. Can I use a plunger on the disposal side?

Yes but if your dishwasher dumps into the disposal it might not do much good since it will just plunge air back to the DW.

If you garbage disposer is backing up, that means the drain - especially the "T" under the sink, where garbage disposer drain and sink drain connect - may be clogged with food. Remove the "T" and clean it. Reinstall. Run lots (lots) of hot water through garbage disposer side of the sink to flush down all food leftovers from that side of the sink

Hope that help

Milo

That and if you use your plunger in the toilet.. its COVERED in Fecal Choliform bacteria among many others. You WILL have to do a heavy duty disinfection to the sink afterwards.

Not a Plumber, and just a thought. How about after blocking dishwasher port, use a shop vac, Sink drains are not accustomed to high pressure, and have seen loosely fitted plastic trap and parts, could see where a plunger may separate the drain components. A
Qualified Plumber would likely make more secure connections.