Is there a verb form of sycophancy or obsequious, a word that would describe someone is conducting sycophancy. Google wasn't of much help.
One is or is not a sycophant. Merriam Webster defines this as: a servile, self-seeking flatterer.
If a person is one, they have engaging in sycophantic behavior for which there are a slew of verbs.
to curry favor, to boot lick, to fawn [over], to kow tow, to brown-nose, to grovel, to suck up to, etc.
Merriam Webster also explains it:
Did You Know? In ancient Greece, sykophantēs meant "slanderer." It derives from two other Greek words, sykon (meaning "fig") and phainein (meaning "to show or reveal"). How did fig revealers become slanderers? One theory has to do with the taxes Greek farmers were required to pay on the figs they brought to market. Apparently, the farmers would sometimes try to avoid making the payments, but squealers—fig revealers—would fink on them, and they would be forced to pay. Another possible source is a sense of the word fig meaning "a gesture or sign of contempt" (as thrusting a thumb between two fingers). In any case, Latin retained the "slanderer" sense when it borrowed a version of sykophantēs, but by the time English speakers in the 16th century borrowed it as sycophant, the squealers had become flatterers.