TL;DR: It depends, but probably not
You might argue that "all 15 samples are within spec, so the supplier should be approved". Not so fast: depending on the parameters of your full production run, the 15 samples may or may not be statistically significant.
There are numerous calculators online to do these calculations; I used this one. A good resource for the actual formulas is online at NIST's Engineering Statistics Handbook or in any undergraduate Statistics textbook.
From the spec, we know that our confidence interval is 10% (two-sided). This means that any parts within 10% of nominal are acceptable.
Next, we need to determine the confidence level. This is usually 95% but sometimes 99%, and represents "how sure" we can be of the result.
The final piece of information we can give if it's known is the population size. In this scenario, this is the total number of parts to be ordered from the vendor over the lifetime of the product/process. Most calculators allow this to be left blank and, if missing, assume a large value, because the effect of the population size decreases as it grows in relation to the sample size.
Assuming a 95% confidence level and our 10% confidence interval, with population left blank, we need a sample size of 96 parts in order to have a statistically significant result. Increasing to a confidence level of 99% requires a sample size of 166 parts.
So, for industry-standard confidence levels, we cannot conclude that the vendor be approved based on the initial sample of 15 parts alone.
Wait a minute—the entire FAIR run was within spec! Well yes, but what's to say the next 15 won't be out of spec? We don't know—that's why we have statistics! :-)
Well, under what circumstances is our sample significant?
Just for illustration, I entered population values until I found a statistically-significant result at $n=15$: for 95% confidence level, $n=15$ samples would be significant only if the population (total production run) is 18 parts! For a confidence level of 99%, the situation is even worse: significance only if the production run is 16 parts!
The calculations above assume that the process follows a normal distribution and that the sample is representative of the population. In practice, both of these assumptions may be inaccurate.