Thanks @maximumsoftware and @approveme!
I think I made it work by implementing two forms:
1o Form: The user fills the form and it goes to his own email address;
2o Form: The user uploads the filled and signed form and sends it to my clients email address.
Either way if the client is not happy with this solution I think I’ll probably go with the Contact Form 7 Digital Signature add-on.
Thanks a lot for all your help!