Edit: I'm going to edit as many of these in to the first post as I can. Thanks for your input so far, this is super helpful to me and I hope others! The remainder of the info given on any specific place will be in your posts below.
I'll try to keep this list edited and updated.
Disclaimer: The idea here is not to discuss your onboard shower idea. I love creative van shower ideas, but I am trying to compile a list of 'real' ones for ripe travelers.
We just returned from about 25 days on the road, and even though I have a shower setup, it's nowhere as good as a bonafide, roomy, steamy hot shower every couple days. My clan and I have been to every corner of the old U.S. of A. in the last 5 years or so, and we have found some great places to get a shower on the cheap, without once paying $12 for a foot fungus at a truck stop. This list is primarily an idea to compile lesser known showers in popular places such as National Parks and near the typical Sportsmobiler type stompin' grounds. It is not an idea to list every known campground and whether or not it has a shower. As most know, the truly wonderful, off the beaten path places are going to have a pit toilet at best usually. That said, every time we go out we are pleasantly surprised at the 'hidden' showers.
For a list of good campgrounds that aren't commercial try the U.S. Campgrounds Project App (or website). It's an iOS app but may be on Android too, I don't know. Under the info section on every campground there is a section for showers or not, and it has yet to be wrong for us.
Feel free to PM me if you want me to add something, or just add it yourself.
Glacier NP area: If you are on the Many Glacier (east side) side, there are token showers behind the Swiftcurrent Motor Lodge, available to anyone. You have to buy tokens at the camp store.
If you are near Apgar on the west side, there are showers in the A-loop of Fish Creek CG, available to campers only.
Grand Teton NP area: Colter Bay CG is the only well known place with pay showers, I don't remember the amount, but they were busy and dirty. Most other CGs there do NOT, however there is a backpacker/hiker hostel about halfway between Jackson and Moran junction that has very nice showers for $2 IIRC, available to anyone. I will update this one. I have the info somewhere.
Great Basin NP area: Just up the road from the tiny town of Baker, NV. There are no showers in the park but a very nice one at the back of the Sinclair gas station in Baker. It's easy to find as it's the only gas station in town. In fact, it and two restaurants are the town. $2.50 unlimited time on the honor system with a payment box in the parking lot.
Jackson Hole, WY: In Jackson WY......the community rec center, right in downtown Jackson. For no additional fee you can use the pool and/or hot tub. If I recall correctly the cost was about $6.
Mammoth, California area: At the Convict Lake Resort, at the Rock Creek Lodge a mile or so before you get to Rock Creek Lake and at the Rock Creek Lake Resort at the lake, both on the road out of Tom's Place.
Moab/Canyonlands Maze District area: Good luck. I will research and see if there are any at Dead Horse Lake SP. I meant to look and forgot. I'm sure there are more in town, hopefully someone can add a couple.
Moab. UT: On 191 (Coming in to town from the North/I-70 -Archview campground, can purchase a shower if not staying there
Also, Poison Spider Bicycles has pay showers. Also, there are several pay/commercial campgrounds that will let you buy a shower w/o staying there. There's also a rec center w/ pool.
Ouray, Colorado (This is near the Alpine Loop and tons of OHV trails in South Central Colorado): Hot Springs at Ouray. The sign at the front said something like "Hot Pool - $12, Showers -$3" (camper rate)
Point Arena CA (This is on Hwy 1 in Northern Cali): On the pier at point arena ,ca. pay showers. tokens available at the bar.
Sequoia NP area: There are pay showers at Lodgepole area, behind the camp store but not inside the campground, available to anyone.
Sierras, East side: link from YVRR to a list from climber.org
Sonora Pass area: Pinecrest Resort has pay showers. Buy tokens at the camp store just inside the resort area on the left. 2.50 per 8 or 10 minutes, available to anyone.
Thermopolis WY (100 or so miles EAST of Yellowstone/Tetons): you can't beat the FREE state hot spring pool. A 20 minute soak is free and showers are included