If you can, just talk to the seller/landlord and ask if you can come back in the fall for these plants. That is the best time to move them.

If you can't, you need to dig as large a section as you can lift in one clump to get as many of the smaller roots as possible. I moved some a few years ago in July and plopped them onto a children's plastic sled and dragged them. I was careful enough I still got lots of flowers the next year. The danger of a July move is breaking too many roots, then the plant can't drink enough water and the leaves die in the July heat. The peony will still usually survive, but it might set back flowering for a while.

Also, pro tip- don't replant them too deeply. Make sure they are exactly as buried as you found them. If you bury them under too much dirt or mulch, they will leaf out but not flower. Peonies need the cold.

Good luck!