Hmmm… we’d have to have a LONG first day (from Washington DC to the Mississippi (r out of the way, we could stay in Chicago the first night then St. Louis the second). Then a long next day to Denver. If we went out of our way, we could crash with someone in Salt Lake City (a relative) or Arizona (friends). Then lots of people we could crash with in California.
So yeah - we’d do some zig-zagging and a couple of very long driving days, but we could just about do it.
Decades ago, we drove cross-country and alternated hotel / camping / friends. The tougher driving days (from the east to Colorado, and from southern California back home) we stayed in motels because they were long days. We mooched off of friends any place we spent more than a day (except Grand Teton). We camped in Colorado / Wyoming and going down the Pacific Coast.
I don’t recommend camping on the long days - it’s a bitch to show up at a campground after dark and have to pitch a tent.