What Do I Need to Extend My 30-Day Visa Exempt Stay in Phuket, Thailand?

That's a 30 day visa exempt entry and to get a 30 day extension you need your passport. There are 70+ immigration offices in thailand and each does things their own way.

There are copy shops near each of the two immigration locations in Phuket, take your passport, proof of where you're staying and a passport sized photo and head over to the immigration office.

Keep in mind they are telling people to WAIT until they have a few days left on their current entry stamp before showing up for the extension because there's so many people there now getting the emergency 30 day covid-19 extension.