I want to submit a speedrun, but I can’t find any stopwatches that don’t require a download since I don’t want any viruses. Is a stopwatch required?

Always check the game and category rules for the game you wish to run, but in most cases you do not need a timer. Downloading from the original website of the software is safe if the software is safe--I can personally vouch for the safety of this page if you want LiveSplit. That said, if you're sure you don't want to download anything, you can always use LiveSplit One, which runs in your browser, no downloads required.