There is (or at least used to be) a trail from the rim overlooking the White House Ruin down into the canyon, which anybody with feet can take. Then you can take what you want, spend all the time you want, and return at your leisure. The elevation change at that point is about 700 feet as I recall--not that bad a climb, with a manageable camera rig.

But don't leave valuables in the car in Canyon de Chelly overlook parking areas, though at the mouth of the canyon (where the tours leave from) is the safest of the locations.

Rick "who has done both the tour and the hike at various times" Denney