Strong gravitational lensing is a powerful tool to provide constraints on galaxy mass distributions and cosmological parameters, such as the Hubble constant, H0. Nevertheless, inference of such parameters from images of lensing systems is not trivial as parameter degeneracies can limit the precision in the measured lens mass and cosmological results. External information on the mass of the lens, in the form of kinematic measurements, is needed to ensure a precise and unbiased inference. Traditionally, such kinematic information has been included in the inference after the image modeling, using spherical Jeans approximations to match the measured velocity dispersion integrated within an aperture. However, as spatially resolved kinematic measurements become available via IFU data, more sophisticated dynamical modeling is necessary. Such kinematic modeling is expensive, and constitutes a computational bottleneck that we aim to overcome with our Stellar Kinematics Neural Network (SKiNN). SKiNN emulates axisymmetric modeling using a neural network, quickly synthesizing from a given mass model a kinematic map that can be compared to the observations to evaluate a likelihood. With a joint lensing plus kinematic framework, this likelihood constrains the mass model at the same time as the imaging data. We show that SKiNN’s emulation of a kinematic map is accurate to a considerably better precision than can be measured (better than 1% in almost all cases). Using SKiNN speeds up the likelihood evaluation by a factor of ~200. This speedup makes dynamical modeling economical, and enables lens modelers to make effective use of modern data quality in the JWST era.