  Recent ALMA measurements have revealed bright OIII 88 micron line emission
from galaxies during the Epoch of Reionization (EoR) at redshifts as large as
$z \sim 9$. We introduce an analytic model to help interpret these and other
upcoming OIII 88 micron measurements. Our approach sums over the emission from
discrete Str$\ddot{\mathrm{o}}$mgren spheres and considers the total volume of
ionized hydrogen in a galaxy of a given star-formation rate. We estimate the
relative volume of doubly-ionized oxygen and ionized hydrogen and its
dependence on the spectrum of ionizing photons. We then calculate the level
populations of OIII ions in different fine-structure states for HII regions of
specified parameters. In this simple model, a galaxy's OIII 88 micron
luminosity is determined by: the typical number density of free electrons in
HII regions ($n_e$), the average metallicity of these regions ($Z$), the rate
of hydrogen ionizing photons emitted ($Q_{\mathrm{HI}}$), and the shape of the
ionizing spectrum. We cross-check our model by comparing it with detailed
CLOUDY calculations, and find that it works to better than 15$\%$ accuracy
across a broad range of parameter space. Applying our model to existing ALMA
data at $z \sim 6-9$, we derive lower bounds on the gas metallicity and upper
bounds on the gas density in the HII regions of these galaxies. These limits
vary considerably from galaxy to galaxy, with the tightest bounds indicating $Z
\gtrsim 0.5 Z_\odot$ and $n_{\mathrm{H}} \lesssim 50$ cm$^{-3}$ at $2-\sigma$
confidence.
