I want to speedrun the DLC Spirit Board for Steve, which literally comes out tomorrow, but I don't have a capture card and same day delivery is not an option. Are we allowed to submit a run that was recorded on a phone? Like I'll make sure the screen is readable and watchable, and I'll still follow the rules.

In the majority of the games you are allowed to record with your phone, as long as the screen footage is readable and watchable as you said. There should be a good reason to disallow such recordings.

Normally I would say that you should ask the moderators of the game you run, to make sure that phone recording is okay. But you say the game is new and not on the site yet, so it has no moderators; In that case you can most probably record with the phone and no one will have a problem.

I would order a capture card and wait assuming you can get one in a few days because you can’t even submit a game request until your account is 7 days old. So you have some time before you’re even allowed to submit games, so you might as well get a card ordered in the meantime.

I’m pretty sure the OP is asking about a new category in Smash Ultimate, not a new game.

I run my games on smartphone and never had a problem with that. I even record them with twitch app or locally with super screen recorder and then I upload it to youtube and submit. It's fine if the footage is readable and you're following the rules of the leaderboard.