I'd be cautious jumping into five axis CAM before having experienced three axis CAM, little bit of running before you can walk type situation. Similarly it's quite an ambitious first build - I don't see why it can't be done, it's just not the easiest way to climb this mountain!

That's also probably why you're not finding commercial machines for the hobby user with large bed 5 axis capability - the high level of programming and understanding the concepts behind the CAM in combination with the mechanical limitations of a large bed hobby priced machine mean it's only ever going to be able to work on material like foam. Fine for you - but not a big market.

I'd suggest starting with a cheap premade three or four axis machine to understand the CAD CAM process first before spending a large amount of time and effort building a five axis machine to realise it isn't quite what you wanted or needed.