  We introduce COMAP-EoR, the next generation of the Carbon Monoxide Mapping
Array Project aimed at extending CO intensity mapping to the Epoch of
Reionization. COMAP-EoR supplements the existing 30 GHz COMAP Pathfinder with
two additional 30 GHz instruments and a new 16 GHz receiver. This combination
of frequencies will be able to simultaneously map CO(1--0) and CO(2--1) at
reionization redshifts ($z\sim5-8$) in addition to providing a significant
boost to the $z\sim3$ sensitivity of the Pathfinder. We examine a set of
existing models of the EoR CO signal, and find power spectra spanning several
orders of magnitude, highlighting our extreme ignorance about this period of
cosmic history and the value of the COMAP-EoR measurement. We carry out the
most detailed forecast to date of an intensity mapping cross-correlation, and
find that five out of the six models we consider yield signal to noise ratios
(S/N) $\gtrsim20$ for COMAP-EoR, with the brightest reaching a S/N above 400.
We show that, for these models, COMAP-EoR can make a detailed measurement of
the cosmic molecular gas history from $z\sim2-8$, as well as probe the
population of faint, star-forming galaxies predicted by these models to be
undetectable by traditional surveys. We show that, for the single model that
does not predict numerous faint emitters, a COMAP-EoR-type measurement is
required to rule out their existence. We briefly explore prospects for a
third-generation Expanded Reionization Array (COMAP-ERA) capable of detecting
the faintest models and characterizing the brightest signals in extreme detail.
