  Constraints on primordial non-Gaussianity (PNG) will shed light on the origin
of primordial fluctuations and the physics of the early universe. The intensity
mapping technique is a promising probe of structure formation on large scales;
at high redshifts, it can provide complementary information to other
cosmological probes. We explore the potential of future wide-field [CII] and CO
intensity mapping surveys in constraining PNG of the local shape, which induces
a distinct, scale-dependent correction to the line bias. We explore the
parameter space of CO and [CII] survey designs that can achieve the nominal
target of $\sigma(f_{\rm NL}^{\rm loc}) =1$ and further calculate what
constraints such surveys can place on PNG of the equilateral, orthogonal, and
quasi-single field shapes. We further test the dependence of these constraints
on various model assumptions; namely the halo mass function, modeling of line
bias, and correlation between mass and [CII]/CO luminosity. We find that the
ability of CO and [CII] intensity mapping surveys to constrain $f_{\rm NL}^{\rm
loc}$ relies heavily on the spectral (redshift) coverage, requiring at least a
coverage of an octave in frequency, to produce significant results, with the
optimized surveys generally covering between $z\sim[2-8]$. As this redshift
window partially overlaps with 21-cm EoR experiments like the Hydrogen Epoch of
Reionization Array (HERA), we additionally explore the prospects for a
hypothetical set of surveys to place constraints on $\sigma(f_{\rm NL}^{\rm
loc})$ utilizing multi-tracer analysis between the lines of HI, CO, and [CII].
