[Harald Xander]

@ Steven
I do not see a sense in deactivating all other plugin or themes. It is pretty clear: If I deactivate connections the customizer works. If I reactivate it again the customizer does not work. Please understand it the other way round. The theme developer would advice to rather deactivate the connections plugin, because this seems to cause the problem.
Nevertheless, I checked both ways to find out about a theme or a plugin conflict. If a deactivate all plugins (beside the connections plugins) this does not help. I still have no access to the customizer.

So, unfortunately, I did not learn anything.