Recent experiments, at room temperature, have shown that near-field radiative heat transfer (NFRHT) via surface phonon polaritons (SPhPs) exceeds the blackbody limit by several orders of magnitude. Yet, SPhP-mediated NFRHT at cryogenic temperatures remains experimentally unexplored. Here, we probe thermal transport in nanoscale gaps between a silica sphere and a planar silica surface from 77-300 K. These experiments reveal that cryogenic NFRHT has strong contributions from SPhPs and does not follow the T^{3} temperature (T) dependence of far-field thermal radiation. Our modeling based on fluctuational electrodynamics shows that the temperature dependence of NFRHT can be related to the confinement of heat transfer to two narrow frequency ranges and is well accounted for by a simple analytical model. These advances enable detailed NFRHT studies at cryogenic temperatures that are relevant to thermal management and solid-state cooling applications.