Every now and then someone other than the court reporter tries to record a discovery.  Below is a case law reference supporting the position that it is not allowed:

Rassaf v. SNC-Lavalin


If you come across parties not wanting to pay for copy orders or really any issue with a client not wanting to pay for a transcript, check out the Supreme Court Civil Rules at rule 7-2(26) where it says "...transcript may be obtained on payment of the proper fee..."