Does IT need a voice of the customer (VoC) program for internal IT customers?

Absolutely... feedback is a tool for self-efficacy and continuous improvement. And who could be the better to provide the feedback other than the folks receiving the service. It won't matter if it is internal or external...

Absolutely. Voice of the customer programs highlight more than just satisfaction or dissatisfaction with IT services, they can provide insight into how well IT projects/initiatives align with desired business outcomes and can be measured in terms of value delivered. This in turn helps both business and IT redefine strategy with a cascade effect through refining the way in which transformation projects or programs are defined, how requirements are gathered, criteria used in vendor selection... and most beneficially helps to reduce technical debt.

Depending on the size of your organization absolutely yes. For smaller companies it may not be a dedicated role, but it should be a standard process that is regularly used