July and August are Monsoon months. Scary lightning. Maybe some rain.
I used to live up on the Mogollon rim, and the folks there expected the storms to start around July 4.
This is a while back, 1994, but we boondocked, with tents, in the National Forest near the park boundary. We did a dayhike halfway down -- Indian Gardens, and then back up. We started down early in the morning, parking was easy to find. Whe we came back up, the area was full of people, cars, and busses. That was in March.
The National Forest has an office in Williams, where you could see if they still allow dispersed camping. It is on the Kaibab National Forest, both south and north rim areas.
I returned with some different folks in 2006, and we hiked all the way down, camped, and spent two days at a leisurely pace coming back up. When we got back up to the top, even though it was March again, the place was a zoo. I had to hurry back for work stuff, so stayed in motels. I actually negotiated the price for a room in Tusayan, but it sounded like they kept a herd of elephants in the room above. Going home, the Williams motel guy looked at my gimp and gave me a handicapped room.
I guess I would go there expecting crowds, unless you hike down into the bottom. Even that area looked well used.
I'd still like to go back. There is a lot to see on the rim, and I have not explored that. There are some short hikes on the South Rim. I believe there are ranger led hikes. You can always hike down into the canyon. The Bright Angel trail off the South rim has drinking water along it. Indian Gardens is about 4 miles down and is a halfway point.
You won't be alone. I live in the sticks and found the people watching to be interesting. There are tourists from all over the world there.
You could look into riding the mules. The trail they use is the Bright Angel trail and it is very wide. It was less scary than some of our local trails in the Cascades, but it consists of steps. Some steps were quite high for my short legs. The mules looked bored and well trained.
The north side is said to be less crowded. I have not been there. It is snowed in during March.
You do not need a permit to dayhike in the canyon. You will if you do an overnight down there. The park service sends out a dvd detailing all the dangers of hiking in the canyon if you reserve a hike in advance. Apparently you shouldn't wear flipflops and set off in 100 degree heat packing a cooler of beer.
