How Does an Email Trigger Behave If the Requester Doesn't Have an Email Address?

We have a trigger that is supposed to send an email and sets a tag that it checks so it doesn't run a second time. But let's say that the user doesn't have an email address in Zendesk yet. Does the trigger run and set the tag, or would it abort the entire trigger because there is no email yet? We would like the trigger to run on every ticket save until they have an email address - which may be added later than the initial ticket save. Then once they have an email, send the message and set the tag.

Is this possible?