Include user information in html block
There is a currentUser global variable that is available to all Twig templates.

If there is a currently logged in user, it will return that user's UserModel class.

From there, it's just a matter of:

{% if currentUser %}
    form=http://formsite.com/formname?email={{ currentUser.email }}
{% endif %}


Of course, what's inside the if block will look different depending on how you're generating the embedded code URL.