When we picked up our new-to-us 2011 Greyhawk 31 DS it also came with no spare. We immediately purchased a new 225/75-R16 Michelin LTX MS2 (which was the newer version of the tires on the unit) at the nearest Costco (cheapest and quickest we could find the tire in the area) and carried it as a loose spare until we got home and purchased a rim.
Many people never purchase a rim but rather just carry the loose spare so they ensure they have the right size tire (at a non rip off price) for the repair guy changing it up for them on the side of the road.
Speaking of tires, if you were not already aware of it (the dealer we purchased our unit from was not) make sure you check the tires on your RV. There is a Michelin recall on currently for the tires supplied on many Jayco Greyhawk units produced from 2010 to 2013. The tires on our Greyhawk were included in the recall and we ended up getting 6 new tires (installed) at no cost to us.
Here is a thread talking about the recall and in one of the posts about half way down there is a link to the Michelin US site that explains the recall :
Jayco Grayhawk Tire Recall
If your Greyhwak treats you half as well as ours has so far you'll absolutely love it !