We've camped at Webber for several years now and we make it an annual trip for so many great reasons. The lake itself is wonderful. The water can be on the cool side but not so much that you don't want to swim in it in August for sure. It's shallow through out most go the lake and much of it is a gravel bottom so water shoes are recommended.
There are two campgrounds - the old Northside campground and the new Westside campground. The Northside has more campsites with various views. The layouts can be sorta hodge podgie - it's not like a state campground where sites are clearly marked etc, but almost all are nice at both campgrounds. The north site has better views of the mountains overlooking the lake while the sites at the west one have better access to the lake. They do allow you to scavenge for firewood in the area - there's a lot of dead and down in the nearby forest so just drive around and pick up what you need. In June and into July the mosquitos can be horrific - I've heard of people leaving as they can be so bad!
They open up reservations usually in March and the website can be finicky and a few letters to ask them to improve it haven't helped. The sites go fast. Their cancelation policy is draconian so make sure you're able to go because getting a refund may not be possible.
The best improvement in the past couple of years is that they now have drinking water. There is no dump station or hookups. There is no cell service either although if you have 4 wheel drive you could drive up to one of the local mountain tops and get service there.
They do rent out one of the small cabins and also there's a big house that sleeps 10 - you have to rent it for 6 days mandatory. It has solar, generator, hot showers, bathrooms, etc., and books out very fast. We're going to be there for the first time in September so I'll try to report about it then.
Kayaking on the lake is the best but you need to sign the release that says you have cleaned your kayak of any possible mussels etc - there's a serious epidemic happening now so please make sure your boats are clean.
Nearby Webber Falls has some great swimming holes although the hike is on very unimproved trails and a bit steep so this is not for those folks who have walking issues.
The Tahoe Truckee Land trust owns the property. The camp hosts are good although there are new hosts this year but my interactions on the phone have been great.