For some reason my Minecraft after 1.20 will not load vanilla version so my only option to play Minecraft is lunar client but I do not know if that is allowed or not? Can someone help.

i don't think so cuz it has optifine on is you multimc or prism launcher not able to work?

My launcher will not download

No, you are not allowed to use Lunar for speedrunning. You said that minecraft 1.20 won't load, but almost every speedrun is done on version 1.16.1, since it is by far the most optimal version to play on, make sure to use this one