Not aware of a specific (PC) program that will receive WX alerts (presumably from NOAA via radio or internet) and send them to a printer.
However, there are several sort of workarounds:
1. This might not fit, but there are marine (and coastal) WX broadcasts called Weatherfax and Navtex, from NOAA in the US, met bureaus in other countries, which can be received by specialized devices and can have either an internal printer or an external one. They're pretty expensive, Furuno makes some of them. There are marine satellite services too, very expensive.
2. NOAA and other WX services have email weather alert services that can be set up to send notifications. Some email clients can then be set up to filter such incoming messages and send them to a printer. Probably your best option.
3. Without a printer, there is good old NOAA Weather Radio, which gives audio and visual alerts on the receiving radios.
https://en.wikipedia.org/wiki/Radiofaxhttps://en.wikipedia.org/wiki/Navtex