Infer position of proton from that of another? I'm a computer science student, so please be patient with my simple understanding of physics.
I heard that if we have a proton, then it does not have a definite position until we measure it.
Suppose we have two protons, A and B, and we put them near each other. We then perform a sequence of measurements of the position of A, and plot a graph of these positions to find the acceleration of A. We can then solve Coulomb's Law to find proton B's position.
But shouldn't proton B's position be indefinite? Are we affecting proton B by measuring proton A? Or is Coulomb's Law too bad of an approximation for this experiment, and the correct quantum mechanical law will give us an indefinite answer for the position of B?
 A: Coulomb's potential can be used in two different frameworks the classical and the quantum mechanical.
In the classical the protons are modeled as points in space with a repulsive 1/r potential.
In the quantum mechanical the 1/r potential is used in a quantum mechanical differential equation which  gives the probability loci of finding the protons at specific (x,y,z) at time t. These probability loci are determined in quantum dynamical dimensions, for the location of the photons very small with respect with the classical point solution, obeying the Heisenberg uncertaitny principle, HUP. 

But shouldn't proton B's position be indefinite? 

It is indefinite within the locus of its quantum mechanical solution.

Are we affecting proton B by measuring proton A?

The effect of measuring in classical dimensions is so  very very small (HUP) it can be ignored in measuring at distances as the ones in particle detectors . Classical   physics mathematical models  emerge from the quantum mechanical mathematics,  in a consistent manner.

Or is Coulomb's Law too bad of an approximation for this experiment

For large distances between the protons , large  with respect to the HUP, it is fine

and the correct quantum mechanical law will give us an indefinite answer for the position of B?

indefinite in a very small locus , not relevant to classical solutions.
If your experiment were one of scattering a proton on a proton, then the quantum mechanical  probable directions  would follow the probability distributions given by the quantum mechanical equations, not the classical ones. This is what high energy scattering experiments are about, determining the probabilities of interaction and fitting them with the appropriate quantum mechanical models. The QM interactions are important in the locus where the two QM particles scatter, locus commensurate with the HUP. Once outside the locus the particles leave a track and can be treated classically in the measurements with the detectors.
A: Some remarks on how we measure.

I heard that if we have a proton, then it does not have a definite position until we measure it.

Imagine you're behind a wall, I can't see you. I have a basketball and I've written down the angle and speed of the ball. I do my calculations well, calculate the gravitational field, maybe the drag, the humidity, ..., I throw ball on ball. You, not very patient, start dancing around the point you choose for our experiment.
One minute I was hitting you with the ball. You scream. In that moment, my ignorance (nonknowledge) of where you are, collapses, and I know your position. More or less, because I hit you in the head, my calculations are wrong. I've calculated, you're a little behind the point you reached. If I hit you on the feet, my calculated point is too close to mine.
I think you got it. Every measuring instrument has its uncertainty.
The smallest things we can measure a proton are what? Electrons? Photons? Another proton? You chose the proton and got Anna's answer.
Think of a laser trap or optical tweezers. Photons have a pulse and laser beams, better together with an electric field, could theoretically keep a proton in equilibrium around a certain point. Perhaps it will be possible to disturb the system and calculate the position of the proton better than with any other method.
And then what? There will be a next check of Heisenberg's uncertainty principle.
You know the resolution of microscopes has gone beyond what was previously claimed? There is a wonderful article "Light microscopy below the Abbe limit", unfortunately in German and even without abstracts in English.
But your thinking is good. If you don't measure the proton directly, but within a system of dependent particles, you could disturb the system and take measurements of the whole system and then deduce the constituents.
