You don't have to have one. It is nice though since everything on the island is a bit spread out. Also, driving is on the left side on the road. If you don't rent a jeep, there are plenty of safari taxis that you can get in Cruz Bay or at the beaches. If you plan on doing this, definitely book it in advance.
Fly into St. Thomas (STT). Then take a ferry to St. John.
You do not need a passport for St. Thomas or St. John. If you plan on exploring the British Virgin Islands, you will need a passport.
Definitely bring several bathing suits and sunscreen. Being a few nice outfits for wedding festivities/dinners. I'd also bring tennis shoes if you plan on hiking. There are several very cool hikes. 2/3 of the island is a national park and can't be developed.