You can do it without a walking foot. It depends somewhat on the machine, as some machines are better than others. No matter what the machine, I highly recommend *heavily* starching both the backing and top before starting. Starch stabilizes the fabric so it is less likely to pucker as you work.
If the quilt is already sandwiched, lay it on the floor on a large flat sheet and do several layers of spray starch on each side, drying layers in-between. A fan helps speed drying of the starch.