That looks like it doesn't have any local-first functionality? Their API documentation only talks about how to retrieve data from their server, even if you are just trying to get the data from your own station.
That said, the point of this system is microcell climate computation, and it works best when part of the public grid of sampling points.
EDIT: I was eventually able to find this: https://apidocs.tempestwx.com/reference/tempest-udp-broadcas...
So it is possible to get the data locally, but it seems like they are pretty strongly discouraging its use.