Permitting for Yosemite is far from impossible. I have gotten last minute solo walk-up permits on many occasions. On a Tuesday after school is back in session, it’s possible every trailhead will have something available. If you’re called there, I would go for it.
That said, there is amazing hiking in the Sierras much closer to Las Vegas. Some excellent (but popular) sections of the JMT to do would be the Rae Lakes Loop in Kings Canyon, starting at Onion Valley (52 miles), or the North Lake to South Lake section through Evolution Basin outside of Bishop (57 miles). That second one would require a hitchhike or an arranged shuttle, which I found pretty easy when I did it. Other sections of the JMT would be possible as well, but might be a longer or more difficult hitch.
If you do decide to go to Yosemite, you can do a couple excellent loops out of Tuolumne Meadows. First, you could go south, heading over Tuolumne Pass to Little Yosemite Valley (with an option for Half Dome) and back over Clouds Rest and Cathedral Pass (52 miles). You could also head down the Grand Canyon of the Tuolumne and then loop over Ten Lakes Pass back to Tuolumne (60 miles). Don’t worry about permits in Yosemite; you can start both of those loops at several different trailheads, so even if one is full for the day, you’ll get another.
I wouldn’t go to Death Valley or Joshua Tree right now — too hot. If you’re going for the “really spectacular” head to the late summer Sierras.
