Are bond angles constant? I'd imagine the answer to the question posed in the title is no - bond angles are not constant. Covalent bonds are not infinitely rigid; hence IR as a spectroscopic technique. 

If so, how can we accurately determine bond angles? What margin of error might we expect from an experimentally measured bond angle?
No, bond angles are not constant. For example, the low-energy IR absorption of water (at $1595 \:\mathrm{cm^{-1}}$) is from the H-O-H bend

There are a few techniques to measure bond angles. For polar compounds, microwave rotational spectroscopy can often be used on small molecules in the gas phase.

I'd guess though, that most bond angles are measured from crystal structures using X-ray diffraction.

Keep in mind that spectroscopic techniques always have a timescale associated with them. So any experimental measurement is time-averaged. In the case of bond angles, while the angle will change, the average value is the equilibrium - at some points, the vibration will create a larger bond angle, and at some points in time, the bond angle will be smaller.