Scott, Welcome to Permies!
My first question was, what's kind of hay?
When we planted our orchard (500 trees over a 3 years, cider trees are hard to source), we just augered 12" wide holes, no effort to remove the mixed grass/alfala beyond the hole. We chop and drop the alleys a few times over the summer. We mixed a pound of rock phosphate and a pound of Azomite into each planting hole, per Michael Phillips recommendation. Also, everything inoculated with Bio-Organics mycorrhizal root dip. Our potassium levels measured low, but I haven't done anything to deal with that yet. I do use Phillips holistic spraying regime in the spring which has kelp in it, a good source of potassium. Our trees are growing very well.
We are slowly getting some clovers to take, to go along with the alfalfa for nitrogen. I'm growing out plants for the understory now. It takes a lot of plants to guild all those trees. Our climate is extremely arid, so if it isn't watered, it won't grow, so doing it in stages as we can manage the water resource. BTW, my observations conflict with the standard advice that grass competes with the trees for moisture. Here it seems the grass protects the soil from evaporation. The ag school guys would disagree

but I know what I see.
Now for your case, in a completely different climate. If it were me, I'd see if I could get the ground lightly harrowed, not enough to kill the hay but to get some ground contact for the seeds, like the implement used to drag manure. Then spread out some clover seeds, wildflower seed can't hurt. Chop and drop the field a few times over the season but otherwise do very little and get my house built and my deer fencing sorted. I'll leave the market gardening part to someone else, but I personally would use the mechanical tillage once to achieve an end. If you could get that done this season and rotate through some cover crops, , that might give you a head start for next season.
Where are you getting your trees?