They both are cognates (it can be easily proved by many etymological sources). The question is : Is it possible to consider VAC as a common root for evacuate and vacuum (we may go further - vacation, vacancy, vacuous etc.).
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Clearly they are related through Latin, from e- and vacare (out of and to empty) and from vacuus (empty), and in Latin the shared morpheme is vac-.
Going further back, these words all ultimately derive from the Proto-Indo-European root *euə- which has the english gloss to lack, want; empty, vacant.
So ultimately this includes many words including: vanish, want, vanity, wane, void, waste and even devastate.