Anybody can fill out a pedigree. The main organization for rabbit breeding in the US is the American Rabbit Breeders Association (arba.net) but they don't have a strict registry like dogs or horses do. You should still join the ARBA. They are the ones that publish the standards for each breed and sanction shows - joining also gives you a nice guidebook and a bimonthly magazine.
Do be warned that MOST of the pedigree templates online are awful. In the rabbit world, a valid pedigree needs the NAME, EAR NUMBER, VARIETY/COLOR, and
WEIGHT for three generations (parents, grandparents, great-grandparents). Along with DOB of the rabbit itself and the address of the original breeder. Optionally they have REGISTRATION NUMBER, GRAND CHAMPION NUMBER, LEGS and WINNINGS. If the pedigree is missing even one of these (eg, missing weight or variety) it's not a valid pedigree. A lot of the pedigree templates online are missing stuff.
Here's an example a pedigree template that has all the correct spots: (note: arba sells a book of pedigree blanks on their site - this is one that somebody scanned into a computer, looks like). Looking through a google search for "rabbit pedigree template", and a lot of them don't have places for some of these values.
If you want to, once a rabbit is 6 months old if they fit the breed standard they can be registered with the ARBA. You need to find an ARBA registrar to examine the rabbit & their pedigree in person though; most people do this at shows. I've got an ARBA registrar license so I look over pedigrees all the time.
