Can I embed a form to my email?

Hello,

For example, under my name signature on my email, I would like to have Client Profile "click here" and when they click it, the form comes up.  

Unfortunately no. The email application/system will most of the time disallow scripts being embedded on the email message. Since the form is also made of script functions, they will not work once you embedded them on email message.

If you want to publish your form via Email message, you can just provide its URL link instead. You can do it directly via the Form Builder.

User guide: How-to-Send-a-form-via-email

Hope this help. Let us know if you need further assistance.