When I read the plot of Carol, on Wikipedia, I'm not so sure she's not at least of French descent.
"Belivet" is a French sounding last name.
And there's the sentence, "Therese's boyfriend, Richard, wants her to go to France with him, hoping they will marry". Why would they go to France to marry if they are not French? Maybe from Louisiana? It's ambiguous.
At the same time, I gather that it's never mentioned if she's French or not in the original novel. Maybe it's implied. But for the movie, they ignored that bit and Rooney Mara didn't have to fake a French accent.