The "Complete" action filters out requests that staff don't want to see anymore (like in the work queue counts or advanced request searches). It does not come into play in any way with what actions are available, so it is possible to complete a Not Supplied request and then go back to it and decide to send it to another lender to get it filled.
However, you cannot fully 'uncomplete' the request once it is completed. So even though it's possible to work on a completed request, it will still be considered completed and will be filtered out of the work queue and advanced request searches. Therefore it may be best to recreate a request once it is Complete, rather than continue trying to work on it.