Edited: Thanks to Simon for correcting me. Evidently, the comma IS necessary. There is a rule that when writing a place name with multiple elements (such as city and state), you need a comma after EACH element, including the last one. I personally find it a bit weird both with and without the comma. I would probably reword the sentence to say something like, "Frank Sinatra collapsed on a stage in Richmond, Virginia, while performing his trademark 'My Way.'" (At least to me, the comma sounds more natural before "while" than before "stage.")

Got it. Thank you Gray!