When there is or in-between two words, the tense is dependent upon the word that comes after or. In other words, the word that is closest to the verb will decide the form of the verb.
Since satellites is a plural form of a satellite, the verb that follows satellites should also be plural. The plural verb in the given options is prove.
As far as the second blank is concerned, you can see Venus is a proper noun; therefore, the verb that comes after Venus should be a singular verb, which in this case is has.
So the complete sentence would become the following:
Telescopes and satellites prove that neither Mercury nor Venus has a moon.
Correct pair: prove:has
