Green wood is definitely the easiest way to go.  With green wood, I think you could make a jig and just hold the limb over heat or steam like Nathan described and then clamp it in your jig until it feels like to much resistance then heat it again and clamp it a little more, rinse and repeat until you have it fully clamped in the jig.  Then I'd leave it clamped up for a week or so until whatever moisture you introduced has a chance to dry out.  Otherwise it's likely to try to straighten out as the moisture leaves the fibers.  If you keep the fibers in tension/compression while it dries, they should retain their length.

If you really want oak and can't find it locally, I'm sure I can find a few trees around here that need a pruning and ship you a few blanks.  Shipping will probably be pretty pricey but I could send you 3 or 4 pieces in a long skinny box.

I think laminations is a lot more work but also probably a superior option.  With a branch, there's always the possibility there are bug tracks or inclusions hidden inside that could eventually lead to a failure.  And with a branch, you will always have the pith in the center which could lead to split ends when it dries out.

You might try posting on some carving forums too.  Those guys/gals may have some they will send you or be able to point you to somewhere that sells them.