The origin of magnetic field sources in the Sun’s dynamo is central to this paper. The Babcock-Leighton dynamo was originally envisaged as a shallow dynamo. The source of the Sun’s magnetism is now generally thought to reside near the base of the convection zone and that these fields rise by buoyancy to initiate sunspots. We reconsider this aspect of the solar dynamo. We do this by considering two surface effects as an alternative to the deep origin of the Sun’s magnetism. They are (1) small-scale convective overturning forming the magnetic carpet of ephemeral regions, and (2) percolation, a process wherein the small structures combine to form larger entities. We discuss these effects, and we develop a numerical percolation model and a set of simplified Leighton-type dynamo equations. The numerical percolation model, initiated with two separate random distributions of unipolar fields, does simulate fields clumping together into larger sunspot-like structures, but does not yet display the bipolar nature of actual sunspot structures. We provide a set of simplified global dynamo equations illustrating the temporal behavior of the current percolation model. With the current model being predominantly illustrative, it is envisaged that more realistic shallow solar dynamo models will be forthcoming. We end by providing three types of observations that may distinguish the percolation model from the deep-seated field origin dynamo models. They are (1) the temporal development of activity centers, (2) the magnetic flux distribution within groups, and (3) velocity flow patterns, near and within active regions. In addition, our modeling suggests that a long-term accounting of the amount of flux in ephemeral regions may lead to long-timescale predictions of solar activity.