Creating a game as a product is a task that requires a team of people with diverse skills, where a game designer plays a central and crucial role.
In a game jam and especially at an amateur level, the truth is that most people tend to work as one-man army. And it's almost impossible for a group to search for a game designer.
What you should do is design your game and then look for a programmer, graphic artist, musician, etc., who are willing to help you bring your idea to life, whether for a game jam or not. But I think it's highly unlikely that it will happen the other way around.
Regarding the GDD (Game Design Document): It's not necessary to have a prototype beforehand, but having one helps, as it allows you to know if a certain idea or mechanic works or not. Creating a good design document helps with game development, but actually creating it takes time, and in a game jam, time is often scarce. Therefore, it's almost impossible for a team to afford to wait for someone to create a GDD before starting to develop the game itself. They're usually created on the fly and in a very simple way, and frankly, they're not usually created at all.