I just got Excitebike for the NES, and I really like it. It's really easy to just pick up and play, and It's a racing game so speedrunning is, like, the easiest thing to do in it. I wanna make a small website where people can just post their times for the different tracks for the game, and show off (and post) their custom tracks. But I don't know anywhere to host this website for free. Any ideas?

There is no for free. In any case you are going to pay with either your personal information or money. Also https://www.speedrun.com/excitebike_nes?h=Selection_A&x=xk9ln6gk

You can host static websites for free at github thanks to github pages. The only con is that the url will be githubusrname.github.io but you can buy a domain and use that (com domain are like 11€ a year)

with the exception of the showcasing custom maps, which can usually be done over discord, all the heavy lifting and time has already been put into hosting level times on this very website https://www.speedrun.com/excitebike_nes/levels