This issue is hosting related. The hosting provider has activated a mod_security rule on the server, that is blocking it’s functionality.
So, to resolve this issue, please contact your hosting support and ask them the following: they should check if your host has any mod_security settings that forbids sending of URLs/content as POST parameters, or any other rule, that could forbid this to happen. If they find such a rule, ask them to remove it. Also, you could describe the issue you are having to them, maybe it will help them to find the mod_security rule that is blocking the plugin’s functionality. To describe your issue, you can tell them the following: “When I submit any kind of content in a custom form using POST method, in a WordPress plugin I am using, I get a 404 page not found/403 forbidden (select the one that matches your case) error on submission.”