Historical weather data with machine learning?
For time-series modelling, one would typically look to LSTMs (or their more recent improvements). Such models are known for their long-time-memory, which is necessary here because of long term seasonality and similar effects.

As for production, there are several methods to serve neural nets, for example Tensorflow Serve.

Specifically about training once in production - you could choose to simply not do it. If your dataset has 10 years of data, and your model will be in production for more than a year, you could think about retraining it (with all the newly collected data + old) periodically, every year or so. It could be completely automated too. 

There are other methods (look for "online training and inference" approaches, which are traditionally in the RL domain). But based upon your description, the extra time you would spend to build such capability doesn't seem to be worth the gain.