Reinstalling the game didn't help, so I figured it was either some sort of VC++ issue, or something left behind in UserData. I deleted files one at a time until I discovered the culprit seemed to be the Reg0.dat file. I launched the game, it recreated the file, and sound worked again. Hooray! So again:
- Close the game.
- Browse to the UserData folder (for the Steam version that's /Steam/steamapps/common/super meat boy/UserData/)
- Delete the Reg0.dat file.
- Relaunch the game, and sound should work again!
If someone feels like updating the Wiki page, that would be awesome.