Quantum spin liquids are a class of magnetic ground states reliant on non-local entanglement. Motivated by recent advances in the control of ultracold polar molecules and the development of dipolar quantum materials, we show that dipolar interactions between S = 1/2 moments stabilize spin liquids on the triangular and kagome lattices. In the latter case, the moments spontaneously break time-reversal, forming a chiral spin liquid with robust edge modes and emergent semions. We propose a simple route toward synthesizing a dipolar Heisenberg antiferromagnet from lattice-trapped polar molecules using only a single pair of rotational states and a constant electric field.