I am certified to coach cheer and tumbling, but we never took more than 5 minutes out of practice to work tumbling and even then it was just to perfect what was in a routine. With the amount of material my hs girls had to learn, I didn't have time or the equipment to train them how to tumble as well. When we competed, the girls all took outside tumbling classes, some still did even when we didn't compete.
As far as necessary equipment, mats are always helpful but again, not every stunt or tumbling skill requires them. If you don't teach safety and spotting techniques it's going to hurt when that girl falls whether you have mats or not. The key is to get them in the mindset to never let the flyer drop or throw a tumbling skill you aren't proficient at.