I'm currently building an iPhone app where the user selects recent text message threads, and chooses which ones to upload to a specific website.
Is it possible to write an app which accesses the users text message database? I heard that Apple is very strict about accessing SMS.
No you cannot acces the SMS message database when using the official SDK.
All apps on iOS are sandboxed and can access files outside of this sandbox, meaning you cannot access any data which is contained in other apps there sandbox. This is also why there is not file browser on iOS.