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 ⁵⁶Ni masses that are broadly consistent with those estimated for type Iax supernovae (0.014–0.478 M_⊙ and ∼0.003–0.183 M_⊙, respectively). To this end, we present a series of models of varying luminosities (−18.4 ≲ M_V ≲ −14.5 mag) with helium abundances accounting for up to ∼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 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_⊙), 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 (≲10⁻³ M_⊙). For the faint type Iax supernova, SN 2010ae, we tentatively identify a small helium abundance from its near-infrared spectrum. 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.