The classical answer (the QM answer would be a lot more work but doesn't change anything qualitatively) is that there is an electromagnetic field if you get close enough to even a neutral atom, and indeed that's why some forms of atomic bonding happen.
However, for a neutral atom this field disappears if you move even a few tens of atomic radii away from the atom. Take the simplest case, hydrogen, with one proton and one neutron. At a point far away from the atom, the electrical fields of the proton are near as no never mind equal in strength (because the distance to the proton is near as no never mind equal to the distance to the electron) and pointing in near as no never mind in opposite directions (opposite charges, but very close to each other) so they pretty much cancel.
But if the atom has gained or lost an electron so that's it ionized, then the magnitudes of the two contributions no longer cancel - there's a net positive or negative charge. An ion with N protons in the nucleus and N-1 electrons around it looks more and more like a point charge of +1 the farther away from it you get.