Yes! It is relatively easy to enable external registrations with Synapse. All you need to do is to change the "enable_registration" option to "true" in your homeserver.yaml file.
One important thing to note, however, is that doing it this way means that the server will not check whether the user is a real human or not. In order to do that, you need to either use CAPTCHA or email for Synapse to send out confirmation emails.