In a hydrogen atom the nucleus only has one proton and no neutrons so the electron to nucleus radio is higher for hydrogen than that for any other atom. Does the orbiting electron induce a wobble on the proton similar to the way a star wobbles due to the orbit of planets. Different forces aside. Larger atoms would be more electrically symmetrical with more electrons.
Perfunctory quantum comment #1: "orbital", not "orbit". There are no little balls moving in a circles in there. These things are quantum objects.
But the sort answer is "yes", the proton has a non-zero momentum distribution that mirrors the electron's. Now, because the proton is nearly 2000 as massive, the proton's position wave-function is nearly 2000 more compact than that of the electron, and its extent can be ignored for many purposes.
However, it matters when comparing the spectroscopy of protium (Hydrogen-1) to that or deuterium or tritium: the changed reduced mass results in very slightly different energy levels.