yep! there's a bunch of tools you can use to make each category
adlib can be made with a number of (usually DOS) tools, faust music creator, adlib tracker ii, reality adlib tracker are but a few
ay-3-8910/ym2149's main tracker is vortex tracker ii
pc engine/turbografx can be made with either huSIC or deflemask
mario paint's an extremely easy one, made so by the lovely 'mariopants'!
midi has many, anvil studio/rosegarden/sheet music notation programs/anything that can export a midi will do
24k tracker module's easiest ones (i think) are openmpt and milkytracker; the stipulation here is the end file has to be below 24 kilobytes
nes/famicom with expansions can be made with famitracker, you can use any expansion here (including pure 2a03 itself)
atari pokey x2 can be made with raster music tracker, has to be set to stereo hack to use the two pokeys
sega genesis/megadrive can be made with either tfm/vgm music maker or deflemask
snes/super famicom is made in openmpt/milkytracker/etc and then converted with either XMSNES or SNESMOD (depending on whether it's .xm or .it), there's a few stipulations such as it being 8 channels or less and certain sample constraints; let us know if you're having trouble with this one!
wildchip is of course anything at all (granted it's blippy)~ you can use this as an allgear format, but you can also impose your own constraints and submit any format not on the list (gameboy, sid, sms, 2x gameboy etc.)