No, most conventional cricket surfaces need a constructed base of some sort. Laying a carpet directly over grass will not provide any bounce at all. There are a couple systems on the market using carpet surfaces laid over special combination pads that do work satisfactorily, but these are intended as temporary pitches rather than permanent installations, offer no adjustment to pace, and are expensive compared to conventional construction.
Flicx wickets can be laid over grass without any base. These wickets are formed from rigid interlocking plastic tiles rather than being a carpet so don't have the look or feel of a usual non-turf wicket. They are ideal for low cost practice and training use.
If you are going to use conventional spiked stumps on your non-turf wicket, an area of the base aggregate needs to be removed during construction and this area packed with heavy soil/clay. This enables the stumps to be pushed into it and stand vertically. Historically a timber frame known as a stump box was made and set into the base to hold the clay. If the base settles the timber can stand proud below the playing surface so generally timber is no longer used, the clay just packed into an area excavated in the base stone. The clay will occasionally need topping up and re packing which can be done carefully through the stump holes.
Using conventional stumps obviously means making holes through the playing surface which in time can tear or leave the playing surface more prone to vandalism at this point. For this reason, it is sometimes preferable to use spring return stumps on a heavy base rather than install a clay pad and use conventional spiked stumps.