One of our clients has a private network that doesn't allow to communicate with Google APIs, so a web service that needs to send push notifications through it is unable to, also devices connected to the network aren't capable of accessing the Play Store to update stuff such an the Google APIs of the device.
What are the URLs and ports that need to be configured to allow all of this communications?
Isn't what was mentioned in the docs what you're looking for?
If your organization has a firewall that restricts the traffic to or from the Internet, you need to configure it to allow connectivity with FCM in order for your Firebase Cloud Messaging client apps to receive messages. The ports to open are: 5228, 5229, and 5230. FCM typically only uses 5228, but it sometimes uses 5229 and 5230. FCM doesn't provide specific IPs, so you should allow your firewall to accept outgoing connections to all IP addresses contained in the IP blocks listed in Google's ASN of 15169.
Also see this possibly helpful post.