"integer multiple" vs. "integral multiple"

As a rule if we have a single-word noun, and if there is a single-word adjective with the same meaning that we get from using the noun as the modifier (especially a directly related one), then while using the noun as a modifier for another noun is acceptable, using the adjective is the more popular decision, and tends to read as more graceful English to many.

In this case however:

1. Integer is better known than integral in this sense.
2. Integral is well-known in other senses, such as "an essential part".
3. Integral has other senses in mathematics, such as the result of integration in calculus, so even those well-aware of the meaning related to integer may have a reason to avoid it.

So while there's no difference in meaning, and you certainly can use integral, there is a good argument in favour of integer here.

I favour integral, but then that's precisely because I'm the sort of person who spends more time on ELU than the mathematics stack-exchange. I'm not the target audience.

Both are accepted. I don't believe there is a difference. As to which is more common, on the arXiv, there are over 7000 hits for "integer multiple" and only around 1500 hits for "integral multiple", so even among mathematicians and physicists, "integer multiple" is more common.

"Integer multiple" seems like a much better term to me, simply because it eliminates any possible confusion that can arise from the word "integral" (like the integral for antiderivatives etc).