You do not have to avoid Flagstaff during monsoon season. You should however use good sense when you choose your campsite to avoid flash flood zones, getting stranded by road closures or stuck in the mud.
I am camped along a well graveled and maintained national forest service road outside of Flagstaff. No issues with with flash flooding here as I am at higher elevation to the north off of hwy 180. Best to be at higher elevation in July and August anyway to keep cooler as well as to avoid flooding. If I go into town on errands I check the weather radar for any intense rain from thunderstorm cells potential. If it looks bad I shelter in place at my casmpsite as directed by the local authorities when they issue flash flood warnings such as the ones the last 2 days.
That flash in Flagstaff flooding was very localized and was totally related to runoff from an area that had a fire in a previous year. That area of the forest is not open for camping as it is in recovery status. Recovery necessary to help prevent this exact type of flash flooding in the future. Alerts for flash flooding were issued for that area before it occured. People were told to stay home and off the roads. But of course they did not and traffic was a mess. The person I have been camping with went into town yesterday, despite the shelter in place order, to get a better cell signal and to pick up packages. He got caught up in the traffic mess caused by road closures in Flagstaff and spent the night doing urban boon docking in an area with some elevation on the north side of town.
Over in Williams Arizona to the East of Flagstaff the Bill Williams NFS camping has been closed for a few years and will remain closed for a number of years to come. The flash flooding danger to Williams is related to that burn which happened several years ago. Fortunately they did not get any problems out of this weeks monsoon event. But the Flash Flood warnings did cover that area as well. The areas by the Grand Canyon were also under flash flood warnings as some of the tourist hiking paths can get flooded.
Stay out of canyons and washes. Watch the weather forecast, sign up for emergency alerts on your phone based on your current location. Look carefully where you camp for signs of strong runoff patterns in the soil. Make sure the road you are on is firm soil and not located in a depression that could accumulate a lot of storm runoff causing you to get stranded.
A lot of people are car camping and totally reliant on solar, often portable panels. Monsoon areas are a poor choice for you. Not enogh solar hours to charge those portable solar generator units as yhey do not charge rapidly. You will be getting damp and unable to dry out clothing or gear inside a small car. Leaving windows open for ventilation is not a viable option in hard rain.
I am better set up for dealing with the rain than a car camper as my build was done in the Pacific NW for camping in rainy weather. Sunny mornings and rainy afternoons mean cooler temperatures at this time of year. So for now I will remain in this area but I will not be camping in a flash flooding prone camp site and not on a foest service road that is prone to closures from flash flooding.Of course in Arizona flash flooding can happen at any time of the year.
Rain is a potential today but so far it has been mostly sunny with a few clouds now and again and a few short minutes of sprinkles of rain. A very pleasant day in mid July with temp in the upper 70s. After living in the Seattle region for 30 years I know I am not going to die from a few, pleasant rainy weeks in summer; extreme heat in the upper 90s is very much harder on my body and my spirit than monsoon season.