Wood stocks have been used on guns for a lot longer than Tupperware has. Take a small cleaning kit and some oil wipes and make sure to clean it if it gets exposed to weather much. Walnut is used because it is pretty (not Impervious) stable. I'd be more concerned with dinging it up and scratching it due to terrain than I would the weather.
Also, isn't the finish on that gun an oil base like tung oil, doubtful a poly is going to bond very well without sanding it down to bare wood. A thin coat of wax in the barrel channel is a good idea.