  Exclusive production of a $\chi_{c1}$ axial vector quarkonia in photon-proton
scattering at high energies requires a $C$-odd $t$-channel exchange. In the
limit of vanishing momentum transfer this occurs either via the exchange of a
photon, the Primakoff process, where the spin of the proton does not change.
For axial-vector meson production, as a consequence of the Landau-Yang theorem,
the Primakoff cross section is finite as $t \to 0$. Alternatively, a $C$-odd
spin dependent Odderon can be exchanged, which involves a spin flip of the
proton. The resulting cross section is related to the square of the gluon
Sivers function or its $\boldsymbol{k}_\perp^2$-moment. Using two models for
the gluon Sivers function from the literature we compute the ratio of Sivers to
Primakoff cross sections and the angular coefficient $\lambda_\theta$ governing
the angular distribution of the $\chi_{c1} \to J/\psi + \gamma$ decay as
functions of $x$. We point out that these observables constrain the magnitude
of the gluon Sivers function at small $x$ which could be accessed in
electron-proton scattering and ultraperipheral proton-proton and nucleus-proton
collisions.
