Is it possible to set an option in the form to not send an auto email?

For example, the user ticks no to the question 'do you want an email copy of your responses'?

Yes. It is possible by setting up a condition on the form using 'Change Email Recipient" option. Here is the guide.

https://www.jotform.com/help/167-How-to-Send-Email-Based-on-User-s-Answer

Please note that the emails with conditions will only be sent if the rule set in the condition is passed.

Hope this information helps!