No rule written in stone. Any Birkin on the shelf is first come, first served. You can be a complete stranger or a loyal customer of several years, as long as you're first to complete the sale.
If you want to order, though, it's totally at the discretion of the SAs. SAs are more likely to offer you the chance to request a special order if you have bought other bags before you place your request, but some people have lucked out and bought relatively inexpensive items first or nothing at all, just hit it off with the right SA.