I know the Ontario Bar collects that kind of information. I am a member of the Ontario Bar myself, and when I complete the declaration as a member of the Bar once a year, I have to provide my identity and specify my linguistic abilities.
As far as universities are concerned, I do not think they collect such data themselves. So the administrative body that governs the legal profession in each province is probably in the best position to collect that kind of information. The umbrella federation of all the provincial bar associations could perhaps ask them to collect that kind of data, but I do not think it's done systematically.
A few years ago, for a study they did, Mr. Power and Mr. Grammond collected data on the number of bilingual judges in each province. Those data may no longer be up to date, but they do exist.
It would actually be up to the bar associations governing the legal profession to collect those data.