Can the center of charge and center of mass of an electron differ in quantum mechanics?

Can the center of charge and center of mass of an electron differ in quantum mechanics?

They can. Particle physics does allow for electrons (and other point particles) to have their centers of mass and charge in different locations, which would give them an intrinsic electric dipole moment. For the electron, this is unsurprisingly known as the electron electric dipole moment (eEDM), and it is an important parameter in various theories.

The basic picture to keep in mind is something like this:

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Now, because of complicated reasons caused by quantum mechanics, this dipole moment (the vector between the center of mass and the center of charge) needs to be aligned with the spin, though the question of which point the dipole moment uses as a reference isn't all that trivial. (Apologies for how technical that second answer is - I raised a bounty to attract more accessible responses but none came.) Still, complications aside, it is a perfectly standard concept.

That said, the presence of a nonzero electron electric dipole moment does have some important consequences, because this eEDM marks a violation of both parity and time-reversal symmetries. This is because the dipole moment $\mathbf d_e$ must be parallel to the spin $\mathbf S$, but the two behave differently under the two symmetries (i.e. $\mathbf d_e$ is a vector while $\mathbf S$ is a pseudovector; $\mathbf d_e$ is time-even while $\mathbf S$ is time-odd) which means that their projection $\mathbf d_e\cdot\mathbf S$ changes sign under both $P$ and $T$ symmetries, and that is only possible if the theory contains those symmetry violations from the outset.

As luck would have it, the Standard Model of particle physics does contain violations of both of those symmetries, coming from the weak interaction, and this means that the SM does predict a nonzero value for the eEDM, which falls in at about $d_e \sim 10^{-40} e\cdot\mathrm m$. For comparison, the proton sizes in at about $10^{-15}\:\mathrm m$, a full 25 orders of magnitude bigger than that separation, which should be a hint at just how small the SM's prediction for the eEDM is (i.e. it is absolutely tiny). Because of this small size, this SM prediction has yet to be measured.

On the other hand, there's multiple theories that extend the Standard Model in various directions, particularly to deal with things like baryogenesis where we observe the universe to have much more asymmetry (say, having much more matter than antimatter) than what the Standard Model predicts. And because things have consequences, those theories $-$ the various variants of supersymmetry, and their competitors $-$ generally predict much larger values for the eEDM than what the SM does: more on the order of $d_e \sim 10^{-30} e\cdot\mathrm m$, which do fall within the range that we can measure.

How do you actually measure them? Basically, by forgetting about high-energy particle colliders (which would need much higher collision energies than they can currently achieve to detect those dipole moments), and turning instead to the precision spectroscopy of atoms and molecules, and how they respond to external electric fields. The main physics at play here is that an electric dipole $\mathbf d$ in the presence of an external electric field $\mathbf E$ acquires an energy $$ U = -\mathbf d\cdot \mathbf E, $$ and this produces a (minuscule) shift in the energies of the various quantum states of the electrons in atoms and molecules, which can then be detected using spectroscopy. (For a basic introduction, see this video; for more technical material see e.g. this talk or this one.)

The bottom line, though, as regards this,

Is there any experimental evidence to support or suspect the center of mass and charge of an electron must coincide?

is that the current experimental results provide bounds for the eEDM, which has been demonstrated to be no larger than $|d_e|<8.7\times 10^{−31}\: e \cdot\mathrm{m}$ (i.e. the current experimental results are consistent with $d_e=0$), but the experimental search continues. We know that there must be some spatial separation between the electron's centers of mass and charge, and there are several huge experimental campaigns currently running to try and measure it, but (as is often the case) the only results so far are constraints on the values that it doesn't have.