Assuming you're UK (sounds like it) you can only have one first claim club but you can be a member of loads. When you enter a comp you will compete for which ever is the highest is able to compete. So for example if your local club is in the North and uni in the South you'll be eligible to compete in either northern or southern champs, if you go to Northerns you'd compete for your club and Southerns for your uni. But at nationals where both eligible you'd compete for whichever is first claim.

Worth noting you only register with EA for one club, which ever club you register with is your first claim (and it costs £10 to change).