Hi Johan !
First, i've ulpoaded the plugin by the normal way and activate it on all the network.
On installation, the plugin create a unique table on database called wp_1_cf7dbplugin_submits. ( if it is not created on installation, it may be created on first submission )
This table store answers of your main blog form's.
The plugin should create a table for each of your blog ... but it doesn't.
I've manually created tables in phpMyAdmin for each of my network site, replacing the number with the ID of each blog.
The problem still that each time i create a new blog, i've to create manually the table.
Hope this will help ! ( sorry for my english, i'll work on it ! promise ! )