Rydberg atoms in optical lattices and tweezers is now a well-established platform for simulating quantum spin systems. However, the role of the atoms' spatial wave function has not been examined in detail experimentally. Here, we show a strong spin-motion coupling emerging from the large variation of the interaction potential over the wave function spread. We observe its clear signature on the ultrafast many-body nanosecond-dynamics of atoms excited to a Rydberg S state, using picosecond pulses, from an unity-filling atomic Mott-insulator. We also propose an approach to tune arbitrarily the strength of the spin-motion coupling relative to the motional energy scale set by trapping potentials. Our work provides a new direction for exploring the dynamics of strongly correlated quantum systems by adding the motional degree of freedom to the Rydberg simulation toolbox.