So, there is a game here to which I have uploaded a speedrun, however, it is not getting accepted, even though it has been months since I have uploaded it, then I checked the mods of that game and sure enough, they have not been active for quite some time.

Normally, in these situations you could just request for a site moderator to replace them with yourself, but... what if I do not have any interest in becoming the moderator of that game and just want to have my run verified?

What do I exactly do? Should I just ask for their replacement as usual and clarify it? Who is going to replace them then?

I do not reckon that a replacement even has to happen, I just want that run verified lol.

You could find an active runner of the game and ask if they want to become mod, and then get them to submit a support ticket. Or you could find a related game. For example, a second part of the game, or of another game in its series. And ask their moderator(s) if they want to be added.

Or you could request for yourself to become a moderator, and then quit after you have verified the runs. So, then board then has no moderators and then someone can easily request to become mod in the future.

Or if the game isn't active, then you could stay as a mod, since you won't have to verify any runs commonly.

Also I don't think site staff will verify your run without adding you as a mod - since I haven't heard of this happening before. Since they wouldn't know if you've cheated or not, or if you've retimed correctly. However, maybe you could ask?

A moderator of a game is also expected to have usually at least 1 run - so it like can't be a random person added.

what if I do not have any interest in becoming the moderator of that game

Nobody wants to be a moderator (or at least nobody should want that, the people who say they do are the ones that worry me the most), but someone's gotta do it. If you can't or won't, and no one else wants too (or no one else runs the game), then you're out of luck