Why can't we influence neutral atoms with electric fields if they have charged constituents? I mean, I know that it's because they're neutral and therefore have a charge of zero, but since atoms have positively charged nuclei and negatively charged electrons, why can't we directly influence the constituents (not the whole) to subject the atom, or at least part of it, to an electric force?
I understand that charges sorta "group up," like 23 protons and 23 electrons with 28 neutrons in a single atom come together like 23(1) + 23(-1) + 28(0) = 0, but what is the extent of this "grouping"? Not exactly sure on the appropriate term for it.
 A: 
Why can't we influence neutral atoms with electric fields if they have
  charged constituents?

We can influence the charged constituents with electric fields. It is just often difficult to see the results of that influence because the constituents are also bound by the strong nuclear force (for nuclei in the nucleus of an atom) or by the electromagnetic force (for the nucleus and its electrons). 
This difficulty is similar to the difficulty of how it is hard to influence a neutral atom by electric fields. However, a high enough energy field will kick out an electron from an atom. The energy in that case often has to be on the order of x-ray energy. Similarly, you can kick apart a nucleus; you just have to use even high energy electromagnetic radiation, e.g., gamma rays and higher.
A: 
why can't we directly influence the constituents (not the whole) to subject the atom, or at least part of it, to an electric force?

I am not sure where you heard this statement, but it is wrong. Electrons inside neutral atoms respond to electric as well as magnetic part of light. Since you mention force, trapping of neutral atoms with an optical tweezer is a common experimental technique.  
EDIT: Not just trapping by light, but laser cooling is common as well. 
