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?
Better than a cover letter is to actually proactively DO something to help them. It'll show them not only that you've researched them, but you're passionate about the startup and worth bringing on.
A man got a job at Square early on for just making them a marketing video on his own (back before they had one). Since you're a web designer, design a stellar 1-pager that's targeting their message to a particular niche. Something they could use on social media or something.
If they're like most startups, they're not interested in reading cover letters. They're interested in passionate individuals who can add value to the organization.