When a charge is moved from one point to another along some equipotential surface, it is true that the work done on the chargeby the electric fieldis zero. Additional work done on the particle over this path depends on how the particle is moved. If the particle is initially moving in the correct direction, then obviously no work is required. This is why the Moon is able to maintain its motion along an (approximate) gravitational equipotential without an army of angels diligently pushing it on it's path.

But if you are moving a particle from one position at rest to a second position at rest, along the equipotential, some external agent supplying work is needed to start and stop the particle. However, you in principle you could make this motion arbitrary slow over arbitrarily long times, so that at any point along the path it's kinetic energy is indistinguishable from zero, hence zero change in kinetic energy along the path, hence zero work necessary.