123 Email signature use and transfer to any other app of information received from Google APIs will adhere to Google API Services User Data Policy, including the Limited Use requirements.
Or in more simple words:
The data that you submit in the Google Form is only stored in your Google Form (and maybe the Google Spreadsheet you connected to your Form) on your Google Drive. The 123 Email signature add-on reads the data from the Google Form that it is installed in. Not from any other Google Form.
After the Google Form user submits a Form and the 123 Email signature add-on is enabled by you, the Form data is processed by the application to merge the HTML template with the field values to your personal Gmail settings. No other processing is done on your data.
You have to activate the Google Form submit trigger to run the script after the submission of the Google Form. For this reason the script needs to be able to run when you are not present.
The 123 Email signature Form add-on is shown as a sidebar in Google Forms.