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I believe the correct term is "vice versa", but occasionally I hear "vice-a-versa" being said. Is there any explanation for that pronunciation?

Vice can have a disyllabic pronunciation because of its Latin origins

As vectory said, the pronunciation with four syllables didn't originate as "vice-a-versa", but as "vi-ce versa", with a non-silent e at the end of vice. Latin doesn't have silent e, and the phrase vice versa comes directly from Latin.

In the "traditional" English pronunciation of Latin, final e's in words like this were pronounced with the vowel found at the end of lily or happy. In an old-fashioned "RP" British English accent, this sound is identified as /ɪ/ (the "ih" sound of "kit"); in other accents, it is identified as /i/ (an unstressed version of the "ee" sound of "fleece"). For example, the e at the end of the word simile, which comes from a Latin adjective, is pronounced this way.

For some reason, vice versa developed a variant pronunciation with /ə/. My guess would be that the phrase was treated as a single word, and so the vowel was reduced more than a word-final vowel would be: for comparison, the word-internal "i" in the word happily is often pronounced as /ə/, even though in most accents it is not usual to pronounce happy with /ə/.

Vice also has a monosyllabic pronunciation

Vice versa also has what seems to be a "spelling pronunciation" where vice is pronounced as a single syllable /vaɪs/. This kind of spelling pronunciation (treating "e" at the end of a word as "silent e") exists for a number of other words or terms from Latin, such as rationale, bona fide(s) and Clostridium difficile.

Aside from spelling pronunciation, another factor that might have contributed to the use of a monosyllabic pronunciation of vice in vice versa might be influence from the French pronunciation of a prefix derived from Latin vice. The OED entry on this prefix says "From the 13th cent. onwards a number of these [compounds] appear in Old French, at first usually with the prefix in the form of vis-, vi-, but latterly assimilated as a rule to the Latin original. [...] The older examples in English, having been taken immediately from French, also present the prefix in the reduced forms vis- (vys, viz-) and vi- (vy-), subsequently replaced by vice- (also in early use vize-) except in viscount n." As far as I know, the prefix vice-, as in vice-chairman, is always pronounced as a monosyllable in English.

According to Longman's Pronunciation Dictionary and the OLD, the standard pronunciation of vice in vice versa is vais in AE but vaisi or, relevant to your question, vaisə in BE. Most likely, the latter variant can also be heard in other parts of the world.

(Note that English often uses the Schwa sound ə for foreign words ending in "e", even though this is generally not correct. For example, German has another Schwa sound. This distinction is missed even by dictionaries and underlies the long-standing controversy regarding the pronunciation of Porsche.)

The e in vice is not silent in Latin. That sound had been pronounced, then rebracketed (like a nadder ~ an adder), and consequently fit into the common *-a-* pattern.