I have done similar jobs in hollow synthetic stocks as well as boring a hole into wooden stocks.

Take it a step further, choose a container profile that suits your storage needs, get some "Great Stuff" or other expanding foam product, insert the container, and fill the stock with foam surrounding the container. I profiled a few AR-15 trapdoor buttplates to fit my shotguns, I keep a bore snake and a tube of oil in the stocks to keep them fresh after a day at the range.

I often use the trapdoor buttplates for hunting licenses, tags, pens, lens-pens, etc etc.