  Recent studies have argued that the progenitor system of type Iax supernovae
must consist of a carbon-oxygen white dwarf accreting from a helium star
companion. Based on existing explosion models invoking the pure deflagration of
carbon-oxygen white dwarfs, we investigate the likelihood of producing spectral
features due to helium in type Iax supernovae. From this scenario, we select
those explosion models producing ejecta and $^{56}$Ni masses that are broadly
consistent with those estimated for type Iax supernovae (0.014 -
0.478~$M_{\odot}$ and $\sim0.003$ - 0.183~$M_{\odot}$, respectively). To this
end, we present a series of models of varying luminosities ($-18.4 \lesssim
M_{\rm{V}} \lesssim -14.5$~mag) with helium abundances accounting for up to
$\sim$36\% of the ejecta mass, and covering a range of epochs beginning a few
days before B$-$band maximum to approximately two weeks after maximum. We find
that the best opportunity for detecting \ion{He}{i} features is at
near-infrared wavelengths, and in the post-maximum spectra of the fainter
members of this class. We show that the optical spectrum of SN~2007J is
potentially consistent with a large helium content (a few 10$^{-2}~M_{\odot}$),
but argue that current models of accretion and material stripping from a
companion struggle to produce compatible scenarios. We also investigate the
presence of helium in all objects with near-infrared spectra. We show that
SNe~2005hk, 2012Z, and 2015H contain either no helium or their helium
abundances are constrained to much lower values
($\lesssim$10$^{-3}~M_{\odot}$). Our results demonstrate the differences in
helium content among type Iax supernovae, perhaps pointing to different
progenitor channels. Either SN~2007J is an outlier in terms of its progenitor
system, or it is not a true member of the type Iax supernova class.
