Can I use that WCF connection to get my computer's external IP
address?
If you want to get your external IP from within the service that sends a request than the answer is no. And it has nothing to do with WCF. Your computer may not even have an external IP because it may not be connected to internet directly. External IP may belong to the router that your computer is connected to. This external IP will be visible to the WCF service you connecting to. So this service can potentially let you know what external IP your request was coming from (if you add new 'MyIP' field in to the data contract).
Your machine -> Request -> External WCF Service
Your machine <- Response(your IP is X.X.X.X) <- External WCF Service
Or you can scrap it from automation.whatismyip.com
Update based on your comment:
... and I want them to be able to get their own IP address and send it
to the server.
They don't know their external IP addresses, but server will know so there is no need to send IPs.