SharePoint 2013 allow anonymous user to submit form public website

we have a public facing SharePoint 2013 Website and wanted to implement a custom form where anonymous users can subscribe for a newsletter. We implemented a custom visual webpart which worked nice in our test environment. After transferring the solution to production unfortunately now clicking the submit button of the form causes a HTTP error 403. Any idea why?

The relevant code of our webpart:

protected void SubscribeBtn_Click(object sender, EventArgs e)
{
    string name = subscriptionName.Value;
    string email = subscriptionMail.Value;

    using (var site = new SPSite(SPContext.Current.Web.Url, SPUserToken.SystemAccount))
    {
        using (var web = site.OpenWeb())
        {
            var subscribeList = web.Lists["Newsletter"];
            web.AllowUnsafeUpdates = true;

            SPListItem item = subscribeList.Items.Add();
            item["Title"] = name;
            item["Email"] = email;
            item.Update();

            web.AllowUnsafeUpdates = false;

            .
            .
            .


        }
    }
}