The keywords you really want to google for are the proton and neutrons' electric dipole moment. Nucleons are not forbidden from having nonzero electric dipole moments, but it's extremely hard for them to do this: nucleons have spin, which is a pseudovector, and the electric dipole moment is a vector quantity which (because of the Wigner-Eckart theorem) would need to lie along the spin, either parallel or antiparallel to it. The problem there is that you need to choose one of the two, which breaks parity invariance, and that typically carries some heavy consequences.
This isn't fatal as such, since the weak force is known not to conserve parity (see e.g. this nice Veritasium explainer for more background), so the symmetry-breaking isn't outside the rules. The problem, though, is that the weak force is weak, and it has only a minor influence on the structure of nucleons.
The upshot of that is that nucleons do have some predicted nonzero electric dipole moments within the Standard Model (so, see e.g. Wikipedia for the neutron's EDM), but they are too small for existing experimental techniques to measure ─ in the case of the neutron, by some five orders of magnitude. This is an interesting research field, because many modifications of the Standard Model predict values of nucleonic EDMs that are plausibly measurable (with, say experimental improvements in sensitivity by only one or two orders of magnitude), so probing those EDMs is a good way to test for physics beyond the Standard Model.
Now, the fact that these EDMs are tiny means that their effect on atomic and molecular structure is pretty much just negligible. For nucleons that's likely to remain the case, but for the electron's EDM the story is a bit different, because it's more directly involved in atomic structure. The electron's EDM is also predicted to be nonzero by the SM but again the prediction is tiny compared with experimental resolution - except that, because atomic-physics experiments have the advantage that optical spectroscopy is extraordinarily precise, you get a lot of leverage in that direction.
And, indeed, since 2011 (paper, eprint) the best experimental resolution on the electron's EDM has been provided by molecular-physics experiments (in the process ruling out certain classes of SM extensions that cannot be ruled out with existing particle-physics experiments). The same might be possible for nucleon EDMs, but I'm not aware of any experimental effort in that direction that's currently ongoing.