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.
A little exposure will not harm your wood stock. Take a cleaning kit as suggested above a clean it after exposure once you return from the hunt.