A few possibilities that I can think of.
Replace the out of the box expire action with a custom action which accepts an input. Then you can run the action to update the property on the VM or push the value into the payload before vRA performs the Expire.
Run an EBS subscription at Expire PRE and use the user action in vRO in theory it would prompt the user and you could have them enter the ticket number and then use the value to update the property on the VM
Run a PRE and POST approval policy on the Expire Action. Have a custom property set on the PRE Approval which the user populates with the ticket number. That value then gets added to the payload and can be used in the POST approval policy workflow to update the ticket status. The PRE Approval would be a standard approval policy, the POST Approval would be tied to an EBS subscription.
Your use case is a bit different to mine as you aren't creating the ticket via a vRO workflow in an EBS subscription.
Attached the action I use to update properties.
I was already thinking about the "user interaction" but just don't like how its being presented (or not presented) to the user in vRA. But I like the approval process idea. I'll give that a go tomorrow.