You have picked one of the busiest weeks, that of the first week in July. First piece of advice is to check online many times a day for sites at Yellowstone and Grand Teton. Cancellations to occur but they disappear quickly. For the Black Hills, do similarly for sites at Custer State Park. As you've probably found, they have their own website. But unless you're incredibly lucky, you won't have much success.
I'm hoping you can get by without hookups. Your unit doesn't have a shower so water isn't important and you will have long hours of daylight that time of year. At many of the places you'll be, you shouldn't need a/c as you'll be out during the day and it cools nicely at night.
Have you used
recreation.gov yet? Scroll down and click on search camping, input the date you want to be someplace, and click on booking types and make sure camping is all that is selected. I found a variety of campsites still available in the Black Hills and Yellowstone area, even for the 4th of July weekend. Sure, no hookups, but having a reservation that time of year is nice especially if you've got a distance to cover and can't get somewhere before noon to grab a first come, first served location.
As noted, there are some first come, first served sites in Yellowstone, but they fill early. For instance we checked a couple of times at Norris last summer as we were exploring and found Norris had filled before 8:00 a.m. both times. Norris is nicely located. Maybe come in the north entrance and drive by Mammoth and Indian Creek to see if you find a spot? Otherwise, I'd take a good look at getting a reservation at one of the CG's around Hebgen Lake NW of West Yellowstone. A bit of a "commute" to get to the park but somewhere with quite a few campsites. There are both reservable and first come, first served sites there. Use recreation.gov and date search when you plan on being there.
As noted at Grand Teton, check out Gros Ventre CG if you are arriving late. But at least at GTNP, a lot of CG's are all first come, first served.
In both Yellowstone and Grand Teton NP's there are showers one can use even if you aren't camped at the specific CG where they are located. Check the NPS websites for info. From our experience, try to avoid 5-8 p.m. as that is the busy time of day and you might waste part of your vacation time just waiting in line.
๐ Although it might be your first experience with your new trailer, you might have to consider boondocking in a few places. At least a Scamp is small and easy to maneuver. Check out
Campendium scroll down to the state you need and click. You'll be able to zoom in to areas to see possible locations.
Ken & Allison
2 Camping Cats (1 diabetic)
1996 4Runner, TRD Supercharger, Edelbrock headers
2007 Fleetwood Arcadia, Honda EU2000i
4 mountain bikes, 1 canoe, 4 tents, 8 sleeping bags, 2 backpacks
(You get the idea!)