Officially registering a name is super awesome, so first of all – rock on you for doing it.
To register a name, you will go through your Kingdom’s College of Heralds. You can usually find it on your kingdom’s website, or by searching “<kingdom name> heralds” in Google. There is a form you fill out with basic information, the name you want, and your documentation.
So yes. Yes you have to have documentation. Even if your name is John or Katherine.
A lot of documentation for “common” names can be found on the SCA College of Arms website.
Heraldry/name registration isn’t really a magical, lost art form that requires animal sacrifice and perfectly drawn chalk diagrams. It’s just rule-following, forms, and documentation.
Happy registering! <3