Well, you can put it out to bid, HERE:
A lot of the East Coast haulers will be tied up running from NY to FL and back for the next 2 months, so you might get a better rate from someone going West to East. I’ve had good ships with Hubbard and Pony Express; there are many, many others.
Key things are: 2 drivers, video monitoring, rest & water stops. On that run you should not need to worry about unloading and hold-overs. Depending on your checkbook, options are usually single straight stall, stall-and-a-half, or loose box. Box stall is worth it for babies, oldies, and anyone who doesn’t tie super-well, but it’ll cost you!
Get your health papers together within 30 days of your projected departure, and make sure any vaccinations are carried out far enough in advance that your horse has full protection when he gets on the van–and off at the new boarding stable. Check with them first as LI may have different vaccination requirements than IL.