I gave permissions to read/edit on Lead and Contacts, I also gave permissions to the user to read on Campaigns and every single custom object and I still don't know why it is not working.

There is also another problem. Before creating the CampaignMember the code looks for a matching contact based on the email. For this specific example, the contact exists so it should find it and create the campaign member attached to it (I am not always looking and creating leads, only if the contact doesn't exist, so I don't want to use WebToLeads). The problem is that it doesn't find it! The query returns an empty list so a lead is finally created.

To see where the problem was I have created a test method that does exactly the same thing (submit a form) and runAs the "Site user" and the weird thing is that it worked!! So it must be something different with the Site, but the user as permissions to see the contact and create the campaignMember in the test method:

static testMethod void myUnitTest() {
List<User> user = [Select Id from User where ProfileId = '00eR0000000IEzw'];
System.assertEquals(user.size(), 1);
System.runAs(user.get(0)){//Member4 is an existing contact in the data Base

WebinarProcessor.updateCampaignWithWebinar('Member4', 'Member4', 'member4@member4.com', 'sv_SE', 'igi', '701R00000002F3o');
}
}