sic'em, egg on, instigate
嗾 centers on urging someone or something to act aggressively: siccing a dog, egging on a person, or instigating trouble.
嗾 combines 口 (mouth) with 族, which likely contributes the sound and may relate to urging or gathering. The exact historical development is uncertain.
The mouth radical 口 beside 族 suggests using words to stir up a group: imagine a mouth shouting commands to sic a dog or egg on a crowd.
For ゾク, picture a zookeeper egging on a lazy animal: zook -> ゾク, and the mouth 口 is urging the creature to move.