Exporting parts for firearms out of the US is controlled by the US ITAR and regulated by the US Dept of State for rifles, handguns and non-sporting shotguns (think short shotguns). US Dept of Commerce regulates the export of parts for shotguns plus the export of sights, mounts and rings.
Certain things (most) cannot be exported out of the US without the proper export paperwork issued by either the US DOS or US DOC.
Strangely enough, shotgun stocks do not require export paperwork from the DOC, if I am not mistaken. You should contact Dept of Commerce directly for a proper answer.
Most US vendors have decided to just not export anything firearm related to Canada (i.e. out of the US) to make it easy for themselves. If shotgun stocks are indeed not regulated for export (handgun and rifle stocks are regulated for export - by Dept of State), you might be able to get the vendor to ship your furniture to someone you know in the US for them to export out of the US to you (i.e. stick them in the mail). Note you can only do this if Dept of Comm confirms shotgun stocks do not require an export permit to export.
Best of luck.