Yes its Possible.
i know this is an old post but this answer is for anyone else looking for the same solution to a similar problem quite recently.
Heres is the simple solution -
First we need to generate an (gmail) email tamplate - this template will contain your signature & the full google form embeded which the customers can submit from that email itself - how you get their attention to fill up the form will be upto your imiganiation.
This template can then be used when emailing any of your customers.
Here are the steps to generate the template (All screenshots attached):
1) Go to the Form that you want your customers to fill up - click on Edit form.
2) Click on Send - enter your own email address.
3) Tick Mark "Embed Form in Email"
4) Open the Email You have Just Received -you will see it contains the form embeded.
5) Click on Reply
6) Delete the subject or Set as per your requirment.
7) Edit your Signature just above the Embeded Form.
8) Once done - click on the 3 tiny dots on the right hand side of the new Email next to the Trash Icon
9) Click on "Templates" - Next Click "Save Draft As Template " Next Click on "Save as New Template" - Set a template name as per your choice.
10) Thast it! The Master Email Template with the Google Form Embeded Signature has been created- Close the New Email Message.
11) To use the template in any new Email - follow step 9 & Click on your Template as per the name you gave it.
This is a one time headache but it gets the job done, if you want to deploy this across multiple users - follow all the steps above & in Step 2 add the users email address instead of your own, help them set it up on their accounts with the very same steps above.
Its quite obvious that you will need to teach the users how to use templates when sending new email meaages to make their lives easier especially if they are not tech savy.