@Carrie
Thanks for clarifying. Now I understand on what you are ruminating. There is no need to worry. I would not buy any ticket in advance given your circumstances. All depends on what time you will be able to board a train after having passed border control, collected your luggage, and cleared customs.
I guess you are arriving by an Austrian Airlines direct flight from Chikago. Given that there is no big departure delay this flight tends to arrive on time or even earlier due to the tailwind across the Atlantic. (I did a short check on flightaware.com.)
Just keep in mind the different options you have. I assume that you will have a device (smart phone, tablet, laptop) to book train tickets (and seat reservations) on the spot.
Direct trains from the airport to Salzburg leave at 8:33am, 9:03am, 10:33am, 11:03am, 12:03pm. It is not likely, although possible, that you can catch the 8:33am train if your flight is early, e.g. arriving at 7:45am. More likely is catching the 9:03am train. If there is time enough prior to departure of the train you could try to book a seat reservation first, and if that is successful buy the train tickets afterwards. If you can't get this done just buy a ticket to Vienna city (€4.30), so you can board the train and decide later.
But you should be able to find seats without a seat reservation as those trains start at Vienna International Airport, i.e. they are empty when they arrive at the platform. The vast majority of passengers boarding the train at the airport will alight at the next stop, i.e. Hauptbahnhof, and vacate a lot of seats. The ride is between 16 and 18 minutes; during that time you should be able to walk through the cars and look for free seats. There is a small display ahead of the windows for every seat, showing its reservation status. That can be blank - good for you -, showing a track section, or showing "last minute reservation". The latter are seats which would be assigned to you if you buy a seat reservation less than several hours prior departure of the train.
Should it really happen that you cannot find available seats, which is hard to imagine, then I would suggest that you go using Westbahn instead of waiting for the next RailJet. The best way to do this is by staying in the RailJet until Wien Meidling train station, that is one stop further than Hauptbahnhof, and from there you take the subway U6 (brown line) directly to Westbahnhof. Buying the tickets online for Westbahn is cheaper than using a ticket vending machine or buying them in the train from the conductor. Westbahn claims that nobody has to stand in their trains, although they offer seat reservations, too.
All this is less complicated as it it might seem now. Buying tickets not in advance gives you more flexibility, but likely not the cheapest price.