Email can't hold a working sign up form, so the workaround is an image of the form that links to the form's landing page. When a recipient clicks the image, the landing page opens and they fill in the real form there.
How to Get an Image and Link for Your Sign Up Form
Click Sign Up Forms.
Open the form you want to add to the email.
Click Preview to open the form's landing page in a new tab.
Take a screenshot of the form on the landing page and save it.
Back on the form's page, copy the link in the Landing Page Link field.
The Landing Page Link field appears once the form has at least one field.
How to Add the Sign Up Form Image to an Email Template
Open the template in the HTML builder.
Drag an image block into the template where the form should appear.
Click the image block, upload the screenshot of the form, and select it.
With the image selected, paste the landing page link into the link field in the settings panel.
Click Save.
Key Notes
A click anywhere on the image opens the form's landing page. Recipients don't type anything into the email itself.
The image doesn't change when the form does. If you add or remove fields, take a new screenshot and replace the image.
To change how the landing page looks, click Landing Page on the form's page.