Hello and welcome to the RTA forums !
I have not stayed in any RV parks between LA and GC although I have seen some good reviews for Blake Ranch RV park just the other side of Kingman which is a good location if you just want to get on to the GC next day. There are other options around Kingman so just Google ?>Kingman RV parks< and see what you come up with. You might want to spend time in Joshua tree NP where I believe there are NP park campgrounds, or a resort at Twenty nine palms, or continue to Lake Havasu, it will depend on how far you want to get on day 1. Laughlin would be another possibility.
We were very happy with our stay at the Oasis which has a free shuttle bus to the Strip, although we felt a $20 cab fare was less hassel than going around the houses dropping off and picking up at other Hotels etc. Circus, Circus on the strip has an RV park but I think it has a bit of a 'parking lot' feel to it, but check it out. Again Google is your friend.
Although you didn't mention it, 'Mather campground' at the Grand canyon is in a lovely wooded setting near to the rim and would need booking asap.
You will have to check your RV rental agreement as they are generally banned during 'summer months' from going into hot desert area's. If any damage was caused or mechanical failure it would make you responsible for the costs. If you are permitted then a good location for night 1 on this leg would be Panamint springs or Lone Pine. From either of these you can decide whether to go South or North on 395 depending on the status of Tioga pass. North and you could stop near Mono lake/Lee vining for the night ready to tackle Tioga in the morning, or South around Lake Isabella. You have a few options open to you and one would be Sequoia NP. If Death valley is off limits and Tioga open you can head to Barstow on I15 and 58 to 395 to Lone pine or head North on 95 and across to Big Pine on266 or up to Tonopah and through Benton on US 6. I would probably take the longer 395 route as you have time to enjoy the scenery.
For Yosemite use one of the 3 'Pines' campgrounds in the heart of the valley floor, wonderful ! Book asap after the booking window opens, sites can sell out within hours of the opening, all info can be found here on the Yosemite NPS site. [Booking window opens tomorrow if there before June 14]
San Fran has a limited amount of RV parks but it is important that you get one close to good transport links, you will not be wanting to take the RV down town ! I have heard of one called 'Candlesticks' that has a free shuttle service but also that is a bit cramped and basic so I would research.
If you are taking the coast Rd between LA and SF you will have to allow at least 2 days and it will be slow going in the RV but there are a few RV parks down the way.
Some links.
You have a very relaxed trip which is nice but you could consider places like Monument valley and/or Zion NP,you might pick up bit's of info from our RV trip in the area and this is a useful site for RV park reviews and this one has a list of campgrounds around Big Sur.