What’s the grammatical role of "for you" in "I am waiting for you"?

Solution 1:

The grammatical role of for you in

1. I am waiting for you.

is that it is an optional argument to the verb. It is not an adjunct. It just happens to be a prepositional argument, not a core argument like subjects and objects are.

It’s also an argument here:

2. She came looking for you.
3. I made a cake for you.

If in #2 for you seems more obviously an argument than it does in #3, remember that #3 is just the dative alternation for the ditransitive two-object version:

4. I made you a cake.

Solution 2:

you is an object in the prepostional phrase "for you"

wait is an intransitive verb so it can not take an object

we can not say I am waiting you

but await is a transitive verb which takes an object

we can say that I am awaiting you