Can I make a call using my webphone if I do not have a number? 

Yes.  You can make an outbound call, however, this will show an anonymous caller-id which lessens the change of the call going through.

Why can't I make or receive calls?

Although all customers have visibility to the webphone, you require a license to make and receives calls. Admins have the ability to purchase a license via their online portal at any time.