Is "I am who(m) God made me" grammatical?

It's grammatical. As per my answer at SAH's question, it's grammaticality is flushed out when one adds what has been (or can be taken to have been) elided, so:

I am who/m God made me to be.

For which I vote for who based on "it sounds better" (the be-all-and-end-all of descriptive linguistics).

Thus

I am who God made me (to be)

is grammatical...

as is the reordered

I am who God made me.

The reason is that traditionally, you can't use who in fused relative clauses; that is, you cannot use who when it figures in two clauses, being the subject (object) of one and the subject (object) of another.

Shakespeare used fused relative clauses:

who steals my purse steals trash.

And they seem to be coming back into use in English today:

I can help who's next.

(Although some people say that this is ungrammatical.)

But traditionally , you can't use who as the pronoun for two overlapping clauses, even though you can use whoever, whatever, or what.

I will do what is right.

There is a good discussion of this on this LanguageLog post, which says much more about it in greater depth than I could.