You will have one regional/state competition in your region/state. In order to attend that regional/state competition, you will typically have to qualify. Your organization will qualify in your particular region/state and would not qualify for a different regional/state competition. So, if I am reading your question correctly, No. Because if your teams qualify for your regional/state competition C, they are not eligible for regional/state competition D. Here in Wisconsin, we have Wisconsin State Championships for VEX IQ, VRC Middle School and VRC High School. Wisconsin teams are not eligible for the Illinois (or Michigan or Minnesota or etc.) regional/state championships.
From what I understand is can multiple teams from the same school enter the same competition.
The answer is yes but I’m not sure if this your question.
Some regions have more than one worlds qualifying competition. Texas, for example, has three. If your organization had a team qualify for the North Regional, and a different team qualify for the South Regional, then they would participate in different qualifying competitions. You cannot, however, participate in a qualifying competition that is outside your home state/region.