"Why am I rich?"
Translation:Vì sao tôi giàu?
You may be referring to vậy because váy means skirt. Adding vậy changes the meaning to "why am I so rich" (rich to this extent, rich like this).
I put: "Tại sao tôi là giàu? and was told this was wrong because it needed to be "Tại sao tôi lại giàu". What is the "lại"? and is there a difference in meaning between "vì sao" and "tại sao"?
In this case giàu is an adjective and the verb là/có is implied, so use it with subject+adjective straight otherwise it will be grammatically wrong. I was told that in comparison to vì sao, tại sao has a more negative undertone (suggesting something happens due to/because of more or less negative reasons), but both are nearly equivalent.
lại means again, so I don't know exactly why DL proposed it for this sentence