NO, they can not get a gate card without a tattoo, and they can't race without a gate card. It is possible, though not terribley likely, for them to have a tattoo without having raced, if they break down between getting the gate card and their first race.
Every horse that races in sanctioned events must be tattooed. I had a Thoroughbred that was definitely trained to race (had it confirmed by the owner/breeder/trainer's vet), but due to unknown reasons, she never made it to the track, and was therefore never tattooed. I believe that its the same in Canada as well. TRPB does tattooing, and its rules apply for both the U.S., and Canada.