What document must service members submit to request permission for an involuntary tour extension?

The correct answer is the Personnel Action Request. This document is specifically designed to initiate various personnel actions, including requesting permission for an involuntary tour extension. When service members need to formally ask for an extension of their tour, they are required to submit this request through a standardized process, which ensures that their needs are properly documented and reviewed by the appropriate authorities.

The other choices do not serve this specific purpose. For instance, the Involuntary Extension Form might seem relevant, but it is not a standard document recognized across branches for this action. The Duty Status Change Form typically pertains to changes in a service member's duty status rather than requesting an extension. Lastly, the Request for Orders is generally used to request official orders for assignments or movement rather than for extending a tour. Therefore, using the Personnel Action Request ensures compliance with military guidelines and proper processing of the request for tour extensions.