I would say a.
1) If he doesn't have either his ID or money, he is in trouble.
2) If he doesn't have his ID or money, he is in trouble.
3) If he doesn't have either of them, he is in trouble.
Do these mean:
a) He needs at least one of the two
b) He needs both of them
There is room for ambiguity.
Missing either of them is a problem. (He needs both)
Not having at least one of them is a problem.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
To me, only 3 is ambiguous.