In the campground, there are multiple pit toilets (no running water), and there will be regular bathrooms and running water in another building a short walk away. The campground also has water spickets.
Alpine (where we live) is a small town with about 800 residents located 40ish minutes south of Jackson. There are lots of outdoor activities to do here (hiking, biking, fishing, trail running, boating, fourwheeling, horseback riding). Greys River Road is probably the best, closest option for hiking, fishing, or relaxing near the river. There is also Palisades Reservoir that is beautiful for adventuring as well! You can easily access Palisades Reservor down our street (Old Alpine Rd). You can rent any type of adventure mobile from Alpine Adventure Rentals (pontoon boats, kayaks, 4wheelers, ATVs, etc.), and they would be a good resource if you wanted to get some info on what to do if you plan to stay in Alpine. Eating options are somewhat limited here, but we do recommend the Taco Bus, Driftwood Pizzeria, Melvin Brewing (amazing beer, pretty good food), and Bull Moose saloon.
This is a tough question! If you are planning to come for just a few days, it would probably be most convenient to stay in Alpine. There are fewer lodging options in Alpine, and there is a Mountain Man festival the same weekend of our wedding, so book soon if you plan to stay here! If you have more time on your hands, we definitely recommend checking out Jackson! The Grand Teton National Park is amazing & just outside of Jackson (15 mins). Lodging will likely be more expensive in Jackson, but we guarantee you won't be disappointed by the scenery. Jackson has a cute downtown square area with good shopping, and there are tons of good restaurants. Some of our favorites are Teton Tiger, King Sushi, The Bird, Hand Fire Pizza, Thai Plate (BYOB!), Dornan's (in GTNP- a rooftop meal here is essential), and Local. Persephone & Genevieve are the best options for breakfast/lunch.
There will be several primitive cabins available that only require you to bring a mattress pad and sleeping bags. Otherwise, we recommend a tent with a rain fly, sleeping pad, sleeping bag, water bottle, rain jacket, extra layers for sleeping, and some sort of light to see at night (could be your phone). There won't be electricity at the campground (or likely service), so a method to charge your phone in your car or through solar energy may be a good idea if you're worried about that. We don't anticipate bears bothering us because we'll have so many people, but it's always a good idea to keep food/toiletries locked in your car (away from your tent). This campground doesn't have bear proof trash cans, so it doesn't seem like an issue at all. You can also rent camping gear at Teton Mountaineering in Jackson, WY, and at a local shop in Alpine. At the wedding, we will provide hand sanitizer & extra toilet paper, but you could always bring your own if you're worried about it.