With your Salesforce Site configured you can set the correct level of permissions for the Guest User permissions by viewing the site detail page and pressing the "Public Access Settings" button. This will open the Profile of the Guest User for the Salesforce Site.
At the top of the profile page hover on the "Enabled Apex Class Access" link and press the "Edit" button as shown in the following image.
On the page that follows you can configure which Apex classes this profile has access to. Find the entry "gams4sf.InboundMessageWebhookRESTService" and add this to the Enabled Apex Classes list and press save as shown in the following image.
Upon pressing "Save" you will be returned to the Profile overview page. This is the only Apex class that access is required for.