I have an Xbox series x and I’m wondering if I’m able to submit runs without a capture card and if so how?

You could always just point a camera at the TV screen to record.

Connect a streaming service and record the gameplay footage right from the console via the sharing options. You can do this for Youtube or Twitch and other services that the platform supports. You can find plenty of information on how to set that up, keep in mind that doing so for a decent stream/recording to be uploaded you should have decent net or else it will suffer some severe quality issues that being said its a nice alternative for those with just consoles and a wired connection to them.