You can create a Google form that does not require signing in to Google (that is, the user can fill in the form anonymously).
Here are the steps.
# Create your form
Go to Google forms website and create a new blank form
Press the new blank form button:
# Add the form Fields
Go to the settings by clicking on the 'settings' icon on the top right
This opens the Forms settings box
Make sure that the "Require Sign in" option is unchecked:
# Test the form in a different browser
You can test whether the form works without any Google Sign in. Open a browser where you never sign in to Google (open Firefox, Opera, Safari where there is no automatic Google Sign in)
In Google forms, click the "Send" button and copy the short link to the form Paste the link in the browser you have opened.
Then see that the form works without Google sign in: