Exactly that... I've done some searching and it looks like a catch-22 situation: To be able to move there you need to do commoning already. But to do commoning, you need live there already. (Hmm... do you? I guess you could live outside New Forest but although your animals would be outdoor most of the time you'd need somewhere for them to shelter...)
Note since some are confused about what a New Forest commoner in the United Kingdom is, here is a quote from the answer by Liam
For those that are interested in what a commoner is, it is basically someone who has the right to gaze sheep, horses, cattle in the New Forest boundary. It was an institution created by William the Conquerer in the 11th Century.