We will be flying from Amsterdam to Vienna, arriving at 9:00AM. We are not able to check into our apartment until 3:00PM. I have asked if there is a place to store our luggage so that we can begin sightseeing but have had no response. Is there luggage storage somewhere or how will we do that if they do not allow us to check in until 3:00PM.
Take the Railjet train from the airport to the Hauptbahnhof. Luggage lockers are in the underground level near the S-Bahn platforms. Railjets go every half hour at 3 and 33 past the hour. From the Hbf, you can get anywhere in Vienna on the S-Bahn or U-Bahn or tram. That is one place anyway. There may be others.
http://hauptbahnhofcity.wien/erkunden/indoor-map/
There is also Westbahnhof...the coin lockers there but location wise Wien Hbf is most likely better suited for you.
Where is the apt located? Closer to Westbahnhof or Wien Hbf?
It kind of depends on where your apartment is located. Can you give an address?