No. Usually all tournaments are open to all, regardless of ECF membership or club affiliation. HOWEVER it may be cheaper for you to play if you are an ECF member.
For instance if you are playing at Hampstead and are an English player, you should hold Gold level of membership with the ECF or above. If you do not, then it will cost you an extra £10 to participate.
If you are playing in any FIDE rated rapid then you should be ECF Silver or above, or it will cost you £8.00 extra to participate.
Check your membership status here - https://ecf.azolve.com/MemberFinder.htm. Ring 01424 775222 to join or visit the ECF website.
You should consider joining your local chess club in order to be eligible to take part in leagues and team events. In this case, you should join the ECF at Bronze level.