Take a look at something like Divekick for UI - you’ll need to get creative with assigning buttons to menu options.
You are allowed to use the mouse buttons - just not mouse position or movement, or the wheel, unless you’re stretching the meaning of “button” and using a click up or click down as a button, but that’s pushing the boundaries of the jam’s spirit.
(Although if you aren’t using mouse position, I’m of the strong personal opinion that using mouse buttons is a mistake.)