There are no designated overnight camping areas in the Sandia Mountains; however, backpacking is allowed. Canyon Young, Sandia Ranger District Trails & Wilderness Program Manager, provided the following guidance in response to an inquiry about a March backpacking trip in the Sandias:
The agency’s general rule for backcountry camping follows the Leave No Trace guideline of not pitching a tent within roughly 200 feet of a trail or water source. The Sandias are a pretty steep & forested range; thus, though there are sufficient suitable places to pitch a backpacking tent up on the mountain, one couldn’t say they are abundant.
FYI, in late March, much of the higher elevations of the Sandias will still be retaining a decent amount of snowpack, potentially making foot travel above 8,500/9,000 feet relatively slow and difficult. And, we often continue get snowstorms into April/May during most years. Not at all discouraging your trip, just things to be aware of as you plan!
Streams are intermittent in the Sandias making carrying sufficient water imperative. The hikers in the very informative video above chose to cache water and food along the trail before embarking on their trek. Canyon Young mentions the problem of snow in the winter and spring months. As mentioned several times in the video, heat can be a serious issue in the summer despite much of the trail being about 9,000 feet.
Of course, all the safety precautions discussed on the Recreation webpage apply when backpacking in the Sandias.