Overall, I can't honestly tell you with absolute certainty, because it's always been sort of a Tuolumne Pass or bust type thing for us....so we've never even really bothered to scout it out. It also depends on whether you want to break it down in terms of 5 miles being measured from the trailhead at Tuolumne or 5 miles from when you actually begin the ascent up from the Rafferty Creek/JMT junction. So, that said, 5 miles of that climb does actually put you at the pass, and there is no camping there, but the best bet prior to that is going to likely be to duck out to trail left somewhere short of the long meadow where you first see Fletcher Peak, better described as being where you get a break from the climb and things sort of level off through the trees. Once you hit that meadow, to the best of my knowledge, since the meadow is delicate, any sites you find would be along the perimeter, up in the talus. You'll undoubtedly find something, but it will likely be a situation wherein you're scroungIng a site in that area where others before you have also run out of steam and had to make do. You are legal in there though:
https://www.nps.gov/yose/planyourvisit/ ... lheads.pdf . This is also assuming that Rafferty Creek still has some flow for camp water, which is not normally the case in mid August.
Since you mention that your trip is a loop, am I correct in assuming that you'll then be headed back down along the Lyell Fork of the Tuolumne on the way out? If so, with the makeup of your party, I would strongly suggest that you reverse that loop and start from the Lyell Canyon end, assuming there are permits available for four of you. Easy enough to abandon one permit and switch right at the permit office if the quota allows. That trail has a substantially less rigorous elevation gain that runs essentially from almost flat to a slight, almost imperceptible, grade along its entire length. At roughly 6 miles in, you also hit Ireland Creek, with plenty of guaranteed water and considerably superior campsites. From there, I have very little doubt that any of you guys would have a huge problem with the remainder of the climb up, and if so, there are multiple options (camping/water) for stopping short between there and Vogelsang, if need be. It also gives you a better chance to acclimatize to the elevation, which is going to probably hit your dad the hardest unless he's had a couple/few days to get used to it before your hike. Seriously, consider it. Why have to plan for potential failure when you can just turn it around and make it work with as much certainty as you're going to get on your way to 10,000 ft?