It there any experimental method to determine the electronic structure of a given amino acid inside a protein?
The two most often used methods used to determine protein structures are X-Ray crystallography and NMR (nuclear magnetic resonance spectroscopy). Both of these allow some deductions of electron configuration, but don't actually measure it directly (and as far as I know there is no method available that can do that, but physicists might know more about that).
X-Ray crystallography essentially detects the electron density of a protein crystal. Usually this density is 'only' used to calculate the structure of the protein, but using good controls and smart math might allow one to also detect charges or specific electron configurations (I haven't checked if anyone ever tried something like this). One general caveat of protein crystallography is that the protein structure/configuration in a crystal may not actually correspond 100% to the state of the protein in its normal environment.
NMR measures the chemical shifts and coupling (quantum) spin of atom nuclei (usually H, N & sometimes C), which is more or less directly depend on its electronic micro-environment. Since structural biology usually doesn't care about things like orbitals, protein NMR data is then used to determine the distance of specific nuclei, which allows derivation of a protein structure.
Since NMR data directly measures atomic properties of single nuclei, I think it is probably the closest to what you are looking for, but asking actual NMR experts will be necessary to figure out whether determining specific orbital configurations could be possible (and even then it would likely be a multi-year interdisciplinary research project).
Another approach, that was mentioned in the comments on the question, would be determining charges on specific amino acids, which could be done using biochemical or spectroscopy assays, if one is only looking at one (or a few) specific amino acids in a protein. These assays need to specifically designed for your protein and amino acid of interest - I can't give specific sources here, but reading up on how people (started to) figure out the chemical mechanisms of enzymes might give you some insights on possible experiments.