I was working with a client recently and they were receiving a 403 error from Zapier. Here is how I fixed up a Zapier 403 error:
Allow Access to and Whitelist the xmlrpc.php file
Some web hosting platforms deny all access to the xmlrpc.php file because it “counteracts a potential brute force attack where an IP address hammers a site for this file” (Source). This file can also prevent remote connections. Make sure that your web hosting providing allows access to the xmlrpc.php file via the webserver (usually Apache or Nginx) configuration settings. You can allow access either to the public or, more securely, to a range of IP addresses (a CIDR). Look at your web hosting provider’s firewall settings and make sure you are white listing the IP addresses that are associated with Zapier.