I noticed there's no section for Universal Paperclip (http://www.decisionproblem.com/paperclips/index2.html) and have been considering running this game (current WR is 1hr 59min I think), is it a game that would be accepted on this site?
To answer the question in your title, yes. There are many browser based games on the site, one notable one being Super Mario 63. Most of the time what players will do (if it's a flash game) is extract the .swf file for the game and load it into a .swf file player. This makes it easier to record and also removes any ads on the site.
Considering I have only run one of those so far, I sure hope they are. Heh. Seriously though you should be fine a lot of people have done this.
In the specific case of Universal Paperclips, you can actually find it here: https://www.speedrun.com/upc
Yea, they are totally allowed, and there is even a Website Speedrunning Discord community: https://discord.gg/WpqZF2y