  Strong emission from doubly ionized oxygen is a beacon for some of the most
intensely star forming galaxies known. JWST enables the search for this beacon
in the early universe with unprecedented sensitivity. In this work, we extend
the study of faint [OIII]$_{5008}$ selected galaxies by an order of magnitude
in line luminosity. We use publicly available UNCOVER DR1 JWST/NIRCam and HST
imaging data of the cluster lensing field, Abell 2744, to identify strong
(rest-frame EW$>500$\AA) [OIII]$_{5008}$ emitters at $z\sim7$ based on excess
F410M flux. We find $N=68$ $z\sim7$ [OIII] candidates, with a subset of $N=33$
that have deep HST coverage required to rule-out lower redshift interlopers
(13.68 arcmin$^2$ with F814W $5\sigma$ depth $>28$ AB). Such strong emission
lines can lead to very red colors that could be misinterpreted as evidence for
old, massive stellar populations, but are shown to be due to emission lines
where we have spectra. Using this deep HST sample and completeness simulations,
which calculate the effective survey volume of the UNCOVER lensing field as a
function of [OIII] luminosity, we derive a new [OIII] luminosity function (LF)
extending to $41.09<\rm{log}_{10}(L/\rm{erg\,s}^{-1})<42.35$ which is an order
of magnitude deeper than previous $z\sim6$ [OIII] LFs based on JWST slitless
spectroscopy. This LF is well fit by a power law with a faint-end slope of
$\alpha=-2.07^{+0.22}_{-0.23}$. There is little or no evolution between this LF
and published [OIII] LFs at redshifts $3\lesssim z\lesssim7$, and no evidence
of a turnover at faint luminosities. The sizes of these extreme [OIII] emitters
are broadly similar to their low redshift counterparts, the green peas. The
luminosity function of [OIII] emitters matches that of Lyman-$\alpha$ at the
bright end, suggesting that many of them should be Lyman-$\alpha$ emitters.
