My game is being really large. I'm trying to get it ready for ifcomp and I'm wondering if it would be better to make it into a zip file containing the quest program also, but I'm unsure what this would look like or how I would do this. Is it possible?

If you publish your game, Quest creates a .quest-file which is a zip container

Does that zipped file contain the quest program also? As in, if someone were to download that file, would they automatically be able to play it on there computer?

Maybe the larger issue I'm having is that my published game is larger the 20mb

Having a .quest file greater than 20 mb will be a problem. There is a bit on this page about that:
http://docs.textadventures.co.uk/quest/publishing.html

Is it possible to zip the game file with the quest program and have it all easily play? I'm not sure how that would work.