From Latin 'advocāre' (to call to) — a doublet of 'advocate,' both from the same verb; to avow is literally to call out publicly.
To assert or confess openly; to declare boldly and publicly; to acknowledge as one's own.
From Middle English avowen (to declare, acknowledge), from Anglo-Norman avouer, from Latin advocare (to call to, to summon, to call as witness), composed of ad- (to, toward) + vocare (to call). Latin vocare derives from PIE *wekw- (to speak, say), which produced an immense family: Latin vox (voice), Greek epos (word, speech — source of "epic"), Sanskrit vak (speech, voice), and Old English woma (noise, tumult). The semantic development runs: "to call as a witness" to "to acknowledge formally
English 'avow' and 'advocate' are doublets — both descend from Latin 'advocāre,' but they entered English through different routes and at different times. 'Avow' came through Old French 'avouer' (which reduced 'advocāre' to 'avouāre' through sound changes), while 'advocate' came more directly from Latin. The word 'vow' (from Old French