Sorry for the delayed response.
I do have firewall rules on my router that prevent the camera from communicating outside my LAN. Would that prevent the data stream connection to the Echo Show in any way?
No, the IP camera should not be exposed outside of your local network.
However, both the Alexa device and IP camera must be on the same internal network and be able to communicate with each other. If they are on separate VLANS or on a “Guest” WiFi network, this could cause an issue.