How to control the purity and quality of fatty acids or their CoA thioesters?
Short answer: Both methods are (technically) suitable but it depends on what impurities you want to detect. But in your case, I would probably suggest mass spectroscopy in most cases.

Long answer:

It depends on how many different compounds you’re getting, how much you trust the company, which impurities you are expecting and how important it is to have exactly the one compound you need.

Mass spectroscopy, ideally coupled with HPLC beforehand, will give a specific retention time for a specific compound, and of course you will get the compound’s mass. That alone is not sufficient for a priori identification! When one detects purity by mass spectroscopy, one would usually have a standard run of one’s desired compounds and compare it to the unknown sample. To identify, compare the retention time (ideally standardised, if the project goes on for more than, say, a year) and the compound masses.

The different fatty acids will have similar, but not identical retention times. The cis-double bond should allow for easy detection via UV absorption (good for quantitising known impurities) and the mass will, of course, differ only by the number of carbon atoms. Impurities that are not fatty acids will have completely different retention times, different UV spectra and different masses which should allow an easy attribution of ‘good peak, bad peak’. If it is not of paramount importance to have only exactly one fatty acid, one can even rest happily with quantitising the amounts of the different ones, too.

Note that it will take some fine-tuning of the HPLC gradient, since most gradients are designed to fit a wide range of polarities with average accuracies, while what you want is a high accuracy in a low polarity range.

Everything I said about HPLC-MS of fatty acids also holds true for the CoA-thioesters, although it is possible that the HPLC solvent hydrolyses the thioester. Work-arounds would be buffered solvents and similar.

$\ce{^1H}$-NMR is best used on pure substances, but can also be used to detect impurities. It should allow for an easy identification on which bond is a double bond by the characteristic shifts and/or multiplets of the two vinylic protons (shift range $6-7~\mathrm{ppm}$). However, you will most likely get unseperable signals at around $1-2~\mathrm{ppm}$ — the aliphatic region where all the saturated carbons’ protons will resonate. Thus it will not be easy to detect saturated impurities, say for example the corresponding unsaturated fatty acid. However, it will be easy to detect other impurities stemming from the synthesis such as remaining reactands (which will depend on the synthetic route).

If you go to the CoA-thioesters, it will be a lot harder to estimate purity just by an NMR spectrum, because CoA has a lot more signals in the $2-5~\mathrm{ppm}$ range. But it is still possible.

NMR would allow the quantification of impurities if you can identify them just by integrating the proton signals and dividing them by the amount of protons giving identical signals.

On a time scale, a typical HPLC-MS run will be around $40~\mathrm{min}$ while a typical proton NMR can be recorded within $5~\mathrm{min}$.

I would suggest NMR in situations where you have a really high number of samples that need to be analysed very timely, where it does not matter if the saturated acid is present or where identifying the stereochemistry and regiochemistry of the double bond is of prime importance. In any case where saturated fatty acids would impede your results, I would strongly suggest HPLC-MS. In either case, I would suggest acquiring pure samples (at least of the fatty acids) first to run a series of standards.