Working on the solution:
...a check that will make sure the email address that they provide
actually exists in the organization...
We were facing a similar situation. Being an educator we have youngsters that quite often misspell their email accounts when filling out their test forms. As a result we would end up sending their grading to the wrong or even a non-existing email.
We are not script savvy so we came up with a workaround.
Instead of having them fill out their emails, we provide those emails as a dropdown list (taken from a class roster) and all they have to do is choose their own.
Assuming the organization is not a couple hundred people (in which case you most definitely need a script), this could very well work in your case as well. If you were to change the droplist to a checkbox, you could even have someone send a greeting card to more than just one member.
You may argue the above answer as the "right" choice. The reason being it may cause a security breach. In addition to having personal data publicly exposed. In that case you could sent the form by email or even have it password protected.
A scripting solution that could, after submitting the form, compare the answer of a specific question to a given list of emails and take an appropriate action (e.g. sent an email to the form creator for a not-listed email) would indeed be very welcomed by us as well.