Vjylku

It is said that atoms with the same number of electrons as protons are electrically neutral, so they have no net charge or net electric field.

A particle with charge cannot exist at the same position and time as another; an electron cannot be positioned at the location of a proton, at any single point in time, without displacing the proton.

Assuming the above is correct, how can a single electron cancel out the entire electric field of a proton? I don't think there is any position a single electron can take, that would result in the entire electric field of the proton being cancelled out - it seems like it will always be only partially cancelled out.

For simplicity, let's look at a single hydrogen atom that we consider to be electrically neutral. It has one proton and one electron, so at any single point in time, there will be a partial net electric field (because the electron will never be in a position where its field can completely cancel out the proton's field), and the electric field from the electron will only cancel out part of the field from the proton. So at this single point in time, there will be a net electric field from the proton. So how can this atom be considered to be electrically neutral, with no net charge or field?

Here is a graphical representation of two sources of electric fields interacting:

As you can see from the image, only part of the (equal but opposite) electric fields produced by both sources are affected by each other. To have the field from one source cancel out the other, completely, we would need to position the sources in the same location, at the same time, which is not possible.

I know that I'm wrong, so please correct me.

If something is 'Electrically neutral' this means that the algebraic sum of its electric charges, however distributed, is zero.

This does not imply that there is no electric field in its vicinity. Plenty of neutral bodies – even, it is believed, the neutron – have electric fields, for just the reason you have pointed out.

It is said that atoms with the same number of electrons as protons are electrically neutral, so they have no net charge or net electric field.

This is a great over-simplification, which I am sure you have already determined (based on why you are asking this question). You can have objects that are polarized where, overall, they have no "net charge", yet the distribution of charge is very important. The example you give is an excellent one of a dipole, where the net charge is $0$, yet $mathbf Eneq 0$ at distances away from the dipole.

Really, the idea of electrically neutral is a macroscopic description meaning that if we look in this general area we will see that the number of positive charges exactly balances our the number of negative charges. However, as we "zoom in" we will find this to not be the case for an "electrically neutral" body, since (neglecting QM) we will have point charges at specific locations, and most of the "charge density" will be $0$ due to no charges being present at all, and then "infinite" (or at least really large) at the locations of the charges.

As it has been noted in Wood's answer, electrical neutrality just means that the algebraic sum of the electric charges is zero. It does not imply anything about the presence of fields.
Notice, that the same situation holds also for electrolytic solutions, so no special role is played by the quantum nature of the charges.

About the fields, I would like to add that it is true that there is nothing forbidding to have non-zero fields in a globally neutral system. However, we should also take into account the observation time. Measurements of electric fields correspond to time average of the fields. Therefore, if a short time measurement on microscopic scale could measure a non-zero field, a time average over macroscopic times could give average macroscopic fields close to zero.