would I need a permit to visit Tibet?

Yes, you need a Tibet Entry Permit to visit Tibet besides China visa, you can't get the permit by yourself, please contact a travel agency and ask it to get the tibet entry permit for you. When you apply for China visa, you don't mention will visit Tibet on the visa application form, then you can obtain a China visa without Tibet Entry Permit. Once you arrive in China, the travel agency sends the Tibett Entry Permit to you.