I suspect the problem is likely on firewall issues with GCP/Vertex.
You may need to create a VPC to open up the port. It looks like Vertex has some docs on modifying the network settings and even this tutorial too for Vertex Notebooks.
Here are the rough steps (correct me if something is different):
Create a Virtual Private Cloud (VPC): Look for "VPC Network" in the Google Cloud Console and create a VPC
Create and add Firewall Rules: "VPC Network" > "Firewall rules" and specify the IP address (0.0.0.0) and the port (e.g., 8888).
Apply the Firewall Rule to your instance.
Just know, that allows traffic from "0.0.0.0/8888" and can pose security risks. You may want to look into more secure authorization controls. At a minimum, Prodigy provides some basic auth with PRODIGY_BASIC_AUTH_USER and PRODIGY_BASIC_AUTH_PASS.