Can a Vanguard have both an Entropy Shield and a real physical shield, and do they stack if so?
No, they don't stack.
The shield has no bulk, but it otherwise functions in all regards as a physical shield of the selected type.
From the rules for shields:
Shield bonuses do not stack with one another, and they do not stack with bonuses gained from cover.
Entropy Shield works just like a physical shield, meaning it provides a Shield bonus to AC. Shield bonuses don't stack, per the rule quoted above.