Probably not a great idea. Raspberries spread and grow like weeds, and can compete hard with the grapes. Pick one, or keep them separate. With very diligent pruning you could probably make it work, but if you're anything like me... not a great idea.

You might get away with a more bush like raspberry, like the new cultivar Raspberry Shortcake which is being marketed for container planting

Raspberries will not out-compete established grapevines, but will spread as mentioned and will make eventual harvesting of both fruits more difficult. 4 feet with a root barrier between the fruits would work.

I've read that some black raspberry varieties don't spread as much.

That is true, blacks don't spread much. Reds definitely spread like wildfire, and I assume yellows probably would too.