"I would assume that it is 8mm film as this is about the inner width of the spool (about 5/16")."
Christopher,
Is that the distance between the flanges of the spools for the finished film, or the distance between the flanges of the camera spool? If it is 'Double 8' then it is almost certainly 16 mm wide going through the camera. There should be an empty spool inside the camera.
The only '8 mm' loads that I'm aware of that came on 'daylight' spools (ie spools with solid sides - no openings except for the spindle) are Double 8 (aka Regular 8, Standard 8) and Double Super 8 (much less common) - all 16 mm wide when going through the camera. Either 25 ft or 100 ft nominal length. Most amateur cameras take the 25 ft spools.
All other '8 mm' loads were in cassettes - either side-by-side coaxial (Kodak's Super 8 format in 50 ft cassettes, and in the 200 ft cassettes if my unreliable memory serves me) or Mickey Mouse ears coplanar (Fuji's Single 8 format in 50 ft). All of those went through the camera 8 mm wide. I mentioned the two film manufacturers as the originators of the format, not as the sole manufacturers of film in that format.
If you look inside the camera and see a gate that is for 16 mm wide film with an opening that is about 5 mm wide then you will have confirmed that you have a Double 8 camera.
Double 8 film, as already mentioned, is double-perforation 16 mm film (in this case 'double-perforation' means perforations beside each edge of the film), but with extra perforations to reduce the frame pitch by half. It can be processed by labs that process the same type of 16 mm film.
You can get
Fomapan R100 B&W Double 8 reversal film from Freestyle. They call it 33 ft long, because that's how much there is on a nominal 25 ft spool.
In WWII 16 mm Kodachrome was used to a much lesser extent than B&W.
Best,
Helen