The short answer
Yes. Quads and buggies are widely offered at Erg Chebbi, usually around the edge of the dunes. Our own desert guide calls them a personal choice: they are fun, but noisy and disruptive to the quiet most people come for. If you want engines, the Agafay near Marrakech is a better place for them.
In more detail
The dunes at Merzouga are busy with activity in season: camel strings, 4x4s, sandboards and quads. A short guided quad run is easy to arrange through most camps. The noise carries a long way across sand, though, and a group of quads can break the silence of a sunset for everyone nearby.
The Agafay, about 40 minutes from Marrakech, is built for it. Its rocky tracks suit quads and buggies, and a sunset quad ride followed by dinner at a camp is one of the most popular evenings out of the city. You keep the Sahara for the camel and the quiet.
Two practical points. Ride only with a guide, wear the helmet and goggles offered, and cover your face against dust. And check your travel insurance before you go, since motorised activities are not always covered. If a quad is on your list at Merzouga, ask for a morning slot away from the sunset hour, when most people want the dunes quiet.