I am a writer and use Twine to make interactive story games. Am I allowed to enter? If I say which songs I used for inspiration or do I have to have the music physically in the game?

I think that's totally fair! Twine are still games. You can even provide links to the Bandcamp page for the album (https://2hac.bandcamp.com/album/entropy-2hac-volume-14) if you wanted people to listen along with the narrative.

This is a great idea, can't wait to see what you come up with :)

Awesome thank you! I have tracks picked out but was not sure if it would be allowed.

You can include music, art and gifs in your twine story, here a link from the documentation:
https://twinery.org/wiki/twine2:add_an_image_movie_sound_effect_or_music

if you are generally interested in interactive fiction, perhaps check out also inform7 and vorple :)

Thank you very much!