Depends on the game. I moderate a relatively active game, and there are a few general rules of thumb for moderators.
They usually need to be longtime members of the community, they need to be very trustworthy, and knowledgeable about the game. They should be active runners with a decent time, and be helpful and active in the game's discord or forums. If you think you meet all of the requirements, DM one of the supermods for that game, and if you are qualified they will add you. Most games have specific requirements other than these, so make sure you are familiar with what is necessary.
There are also rules on the site for who can and can't be a mod. Mods should be at least 16 years old, have at least two socials where they can be directly contacted, be active on the site, have an account that is at least a month old, have a pending/verified run in the game, and represent their community. You can read the full moderation rules here: https://www.speedrun.com/knowledgebase/moderation-rules
Also keep in mind that moderation isn't a fun thing, or a title or a rank. It's a volunteer job, and it can take a lot of time and effort, so don't sign up for it if you don't know what you're getting into. Some people who moderate a lot of games or very popular games can have over 400 pending runs at a time, (although it's pretty extreme, I usually never have more than 10).
I’ll try to answer your question more literally. There are three ways you can be added to a game as a moderator.
- Be added by a current super mod.
- Request a new game to be added to the site, then you automatically become moderator.
- Request for site staff to add you to replace an inactive mod.
jack. 3 months and you added nothing important to the thread