I read all the terms of service but couldn't find an answer to this. If I use the itch io logo and name in my videos, will there be any copyright issues?

You can mention the name and logo of a company to discuss it, as a general rule. Check out the press kit.

(I'm not a lawyer, and this isn't legal advice, but you're probably fine, and thanks for talking about itch.io!)

Wow i did not expect such a quick response .thanks