All tournaments can and do make their own rules about everything, so it is helpful to know what they are before you show up. Most tournaments will allow anyone to shoot, but if the local rule is that the trad class requires shooting off the shelf, it would be a shame to be pushed into the open class and have to compete with the Olympic style shooters with all their cushion plungers, stabilizers, etc., just because you have a stick-on rest. Some tournaments also have different rules for recurves and longbows.
The level of sticking to the rules varies with the personalities of the shoot organizers, and whether the shoot is officially sanctioned by NFAA. At one shoot I was in a few years ago, one of the organizers didn't like it that one of the people in my group was string walking, so he asked for his scorecard and wrote a big DQ across the front of it. He was allowed to continue, but couldn't submit his scorecard for scoring.