I don't have a youtube channel and I can't currently get one, Is there any way to submit a video of the run that the mods can see without the video being from youtube?

The video should ideally be on some streaming site like Twitch for example. But you can check with the mods of the game in question to find out if they have any additional sites they support for submissions.

ok thanks

If you stream on Twitch, you can "highlight" your stream to keep the video online forever.

If you can't stream on Twitch or upload to YouTube, then you could try recording your screen with something like a video camera, phone, etc.

But as Timmiluvs says, check with the community of the game you are submitting to as some have certain rules.