Online petitions are inherently worthless for a great many reasons, and it isn't worth the effort since it won't mean anything. For that matter, the whitehouse.gov petitions are twice as worthless since gun owners generally are too paranoid to sign them. Not that they'd get a favorable response anyways.
If the NFA is essentially tax legislation then sliding an amending clause - that reopens the Registry - into one of the various multiple volume tax bills would seem to be an effective way of progressing the issue.
Why does this keep coming up? It seems one of the worst ways possible to progress the issue, and as Tom Servo notes, the time isn't right. Such a violent push would certainly result in a net loss.
First, it's political suicide for whoever poked that clause in there. Who's willing to sacrifice their career for machine guns? And then how many days would it be until some sharp eyed reporter notices the last minute clause that doesn't point to tax code? There's no way it would stay on the books long enough for anyone to get their stamps.
Currently, the general public, and even probably most gun owners are fairly ignorant of NFA law. Chances are, the public response would go like this:
"What do you mean anyone can own machine guns? What do you mean the Machine Gun Tax is only $200? What do you mean it hasn't changed since 1934? What about the puppies and the kittens? THINK OF THE CHILDREN!"
How do you think the legislative response would go?
Waiting for a crack in the registry is about as good as we can hope for in the meantime.