You probably already read this but according to the FAQ on the official website, you may be able to get tickets the day of travel.
However, it doesn’t sound too promising:
On some of our trips (The Jacobite, Scarborough Spa Express and The Dalesman) limited tickets may be available on the day, which include First Class, the HP style Compartment Carriage (only available on The Jacobite) and Standard Class. These are sold on the Platform direct from the Guard on a first come first serve basis therefore early arrival (approximately an hour before departure) is recommended. These cannot be secured beforehand and there is unfortunately no guarantee of how many, or if any, tickets will be available on any given day. Please also note the 'on the day' tickets are cash only.
Edited to add;
I was intrigued by your question so did a bit of research.
I’ve not taken this train (yet) but you
can take a regular train on this route and you can book your tickets (usually 12 weeks in advance of your travel date) here: www.scotrail.co.uk/
Wishing you a great trip!
P.S. If your itinerary includes Edinburgh please let me know if you’re interested in restaurant/ pub recommendations. My daughter lived there for 5 months during her study abroad semester and she compiled a couple of lists.