I have come across a startup in the very early stages, and I'm interested in working with them. They recently posted a job opening for a web designer and I want to apply. Is there any specific thing I should do regarding applying to a startup? Like a cover letter or something else?
PICK UP THE PHONE AND CALL THEM....
Cover letter? Are you kidding me. The only people who write cover letters are "no one".
If you can find a phone number then call. If not, drive to where they are and meet with them. If you want to work for them bad enough, you'll invest the effort into talking with them face-to-face.