Whelp, I found a $42 board and $10 shipping. I guess I'm willing to pay $52 for a whole board instead of mucking around with ordering a relay, desoldering the old one, soldering in the new one, and possibly still having a malfunctioning control board.
Relay would have cost me $10; $2 for the part, $8 to ship.
Funny how annoyed I get having to open an app on my phone to start the vacuum rather than just push the switch on the handle.
The house also has a hidden vacuum port that triggers the central vac to turn on when you open it. It's meant to sweep the kitchen, open the inlet, and have everything disappear. Would be nice to have that going again.