der betreibende Verein
"Verein" can mean "association", "club", "society", "assembly", "organization", and in some contexts even "outfit" or "charity".
"betreibend" means "operating something". However, this "something" is not the "Verein" itself, but something else (either implied or mentioned).
Getting back to your original sentence,
Die Adresse des Frauenhauses hält der betreibende Verein "Cocon" geheim.
you would normally get the implication that the "Cocon" association is the one operating said "Frauenhaus". A possible translation of the whole sentence could be
The women's shelter's address is kept secret by its operating "Cocon" association.
or (in an active mood)
The "Cocon" association, who is operating the women's shelter, keeps its address a secret.