I'm curious. I believe national-wide weather forecasting by computers was been done since the 1960s. As today's personal workstations are much more powerful than a supercomputer built in the 1990s, and the old models and numerical methods (not the state-of-art ones) are well-known (and have multiple open-source implementations), the calculation should be possible today on an ordinary workstation for educational purpo…
For all but the most extreme configurations, WRF will run on a modern 2-4 core Linux desktop. It will be fairly slow, but it will run.
You can use the raw data from the official model runs published by NOAA as initial and boundary conditions for your model runs: https://nomads.ncep.noaa.gov . One of the coolest and most under appreciated things about NOAA is that they publish everything online for free for everyone.
Or you can just get the raw data from the official runs from the link above and do your own extraction, and maybe post processing if you like.
(Source: PhD in meteorology. Finally made an account when I saw this posted)