There's an Open Plant Hardiness Zones (OPHZ) project on Github where various people have reverse-engineered a pdf of the hardiness zones (pdf, really large) to produce a GIS file (SHP) of the zone boundaries. The ophz-c version is the latest. It's public domain.
You could use GIS software or tools, and a list of the center point of each zip, to find the hardiness zone for each zip. (you're no doubt aware there are no firm boundaries for each ZIP, so this would just be an approximate answer, the latitude and longitude of a street address would be better).
Doesn't even have to be GIS software. You could convert the SHP file to geojson if needed, import it into a geo-aware database such as MongoDB or Postgres, import the zip code list as well, and use the database GIS queries to give you "zip is within .. zone" answers.
Probably a more complex solution than you'd like. Maybe somebody has done these steps already and released the results, but I'm not aware of it.