That you must carry your radio operator’s certificate with you on board the aircraft is a widely spread myth in Canadian aviation. It’s not true! Here are some examples of that claim in the wild:
There are a number of documents that must be on board in order to fly […] I carry a Restricted Radio Operator Certificate, restricted to aviation. (Bits of Paper)
During flight operations in Canada, the following documents must be carried aboard the aircraft […] Pilot radiotelephone operator’s certificate… (Required documents)
So an operating certificate is always needed wherever a Canadian pilot is operating a
radio on a Canadian aircraft […] They have not been inspecting Canadian pilots recently to ensure
that pilots are carrying this licence, but can do so at any time. (The AOPA/COPA Guide to Cross Border Operations, Page 31)
The following must be carried by the pilot […] A Restricted Radiotelephone Operator’s Certificate is only required if you intend to transmit on an aviation-band radio. (Are you legal?)
Also on board must be […] the radio operator’s licence of the pilot […] (From the Ground Up, 27th Revised Edition 1996, p. 100)
What do the regulations actually say?
A person may operate radio apparatus in the aeronautical service, maritime service or amateur radio service only where the person holds an appropriate radio operator certificate as set out in column I of any of items 1 and 3 to 15 of Schedule II. (Radiocommunication Regulations, s 33)
The holder of a radio authorization shall, at the request of an inspector appointed pursuant to the Act, show the radio authorization or a copy thereof to the inspector within 48 hours after the request. (Radiocommunication Regulations, s 38)
You only need to produce the radio operator certificate (or even just a copy!) within 48 hours of a request by an Industry Canada inspector. You do not need to be able to produce the document while exercising the privileges. When Canada wants you to have the document with you, it knows how to say that:
[…] no person shall act as a flight crew member or exercise the privileges of a flight crew permit, licence or rating unless (a) the person holds the appropriate permit, licence or rating […]; and (d) the person can produce the permit, licence or rating, and the certificate, when exercising those privileges. (Canadian Aviation Regulations, s 401.03)
You must be able to produce your pilot licence and medical while exercising their privileges. You don’t need to do this for the radio operator certificate.