I currently have 2 hay suppliers--one has alfalfa/grass hay and the other has straight brome hay--both took my order at haying time & then keep the hay & just deliver it as we need it. Neither asked for payment in advance, not even a down payment--I pay for each load on delivery. This works great, because 1) they have the hay under cover, either in a shed or well tarped, so it's protected from the elements; here we have limited hay storage room so we don't like to have too much hay stacked up. It's not a problem in winter, but in spring, summer & fall we like to have just what we have room for in the hayshed and 2) when you're using in excess of 2400 bales a year that's a big outlay of money to pay for it all at once--much easier on the wallet to pay for it as it's delivered!
Edited to add: In my case, both hay suppliers set their price at haying time, and once I speak for my hay, I get that amount of hay at the same set price all through the course of the year.
If you don't have the hay spoken for, though, and have to go hay shopping in March or April, it's almost certain that the hay will be higher priced than it would have been if you bought in the fall. Some of the guys hang onto their hay until spring, because they know there will be a demand for hay at that time, and so they will be able to ask more money than if they had sold it in the fall. Such isn't always the case of course--in a year where hay is plentiful and then the winter is an easy one, with spring coming early--then there will still be plenty of hay available, and an early spring means cattle (and horses) can go out to pasture early, so in a year like that you might actually be able to buy hay quite cheap in the spring. But, if hay is in short supply to start with, then it's a hard winter with a late spring--demand will be very high and supply will be limited, and in that situation you will pay dearly if you have to buy hay in the spring.