Check the weather. You may need to stake something for a privacy shelter, but I doubt that it will rain. Look for some weather history. Most of the rain comes in mid-summer and mid-winter storms with not much in between. The humidity will be so low in June that you will not see a cloud for weeks.
Although Phantom Ranch is miles away, it does have regular weather reporting that Supai may not have. Meadview and Dolan Springs are tiny villages that may have the same weather as Supai.
There is a Hike Arizona forum that could help with specific knowledge of the campground in Supai after the flash flood. You will have rocks to put on top of your thin stakes.