There’s a whole lot of airports with approaches along that route. Usually time aloft, bathroom and food breaks are going to dictate “the plan” more than worrying about the controllers being willing to allow an approach. Most of the time that’s going to be “practice approach approved, no radar separation services provided” in super busy airspace.
When in the middle of nowhere in quiet airspace (which most of that is) it’ll be “Center, we are on a long training flight and if it would work for you we’d like to shoot the [insert approach here] then fly the missed and continue on to our destination if that’ll work for you”, and (from Center), “Do you want vectors or to fly the full procedure.?” You answer. They respond, “Okay cleared practice RNAV blah blah approach at Blah Blah, on the missed just contact me on this frequency for further flight following once you’re above 3000, we probably won’t hear you until then.”
Or whatever. If you’re landing and it’s VFR they’ll just toss you to CTAF early.
Also depends on if you’re VFR or filing IFR as to how you’ll handle it. If you’re IFR you could file it with airports listed along the way and comments that you’ll be wanting practice approaches along the route at X Y Z airports but many facilities don’t see the comments so you’ll end up just talking to them about the plan anyway.
Don’t overthink it. I’ve done XCs where multiple approaches were added ad hoc along the way just by talking with the Center controller. If all you’re looking for is lots of approaches.
If you’re looking to land somewhere after the approach for a decent lunch or to drain the human sumps, that’s a whole different question about how to choose an airport for a landing during an XC.
Even in and near large airport airspace on that route you’re likely going to be able to get in as many approaches anywhere that you want to if you’re VFR/VMC. Until Chicago. They may or may not be helpful.
No big deal. Your CFI may have a few tricky ones already in mind, too. Or knows where the food is best and who has the free courtesy car to use to get to the BBQ joint in town.
