Supernovae (SNe) that have been multiply imaged by gravitational lensing are rare and powerful probes for cosmology. Each detection is an opportunity to develop the critical tools and methodologies needed as the sample of lensed SNe increases by orders of magnitude with the upcoming Vera C. Rubin Observatory and Nancy Grace Roman Space Telescope. The latest such discovery is of the quadruply imaged Type Ia SN 2022qmx (aka, SN Zwicky) at z = 0.3544. SN Zwicky was discovered by the Zwicky Transient Facility in spatially unresolved data. Here we present follow-up Hubble Space Telescope observations of SN Zwicky, the first from the multicycle LensWatch (www.lenswatch.org) program. We measure photometry for each of the four images of SN Zwicky, which are resolved in three WFC3/UVIS filters (F475W, F625W, and F814W) but unresolved with WFC3/IR F160W, and present an analysis of the lensing system using a variety of independent lens modeling methods. We find consistency between lens-model-predicted time delays (less than or similar to 1 day), and delays estimated with the single epoch of Hubble Space Telescope colors (less than or similar to 3.5 days), including the uncertainty from chromatic microlensing (similar to 1-1.5 days). Our lens models converge to an Einstein radius of theta(E) = 0.168 (+0.009)(-0.005) the smallest yet seen in a lensed SN system. The standard candle nature of SN Zwicky provides magnification estimates independent of the lens modeling that are brighter than predicted by similar to 1.7 (-0.6) (+0.8) mag and similar to 0.9 (-0.6) (+0.8) mag for two of the four images, suggesting significant microlensing and/or additional substructure beyond the flexibility of our image-position mass models.