I think to be on the safe side you should have a printout on letterhead of the shots in chronological order and with details about each shot (live, deactivated, whatever). The shot record should be signed by the doctor.
www.ops125.net/vimages/shared/vnews/stories/4e69309bf1af7/17-18%20grade%20level%20immunization%20requirements.pdf
Summary of Illinois immunization requirements by grade level.
There are minimum intervals between increments in series and for boosters, so watch out for compliance.
The nursing staff in the pediatrician's office will fill in the proper Illinois form using whatever proof of immunisation you bring, and then you can file the red book. The pediatrician will sign off on the record if satisfied with what you bring with you. All Chicago schools, public and private, require the proof of immunisation on the proper form.
They do preliminary vision and hearing testing in all Illinois schools, public and private iirc. If anything is flagged then you take your child for further testing yourself. One of my DCs ended up with glasses as a result of this protocol.
You have to get a dental exam done yourself. It's possible a recent dental exam from a British dentist would be acceptable, but again, letterhead and signature would be needed.
Just bring duvets and your favourite pillows - US beds and sheets come in different sizes from British, and duvets are less a thing here than they are in the UK.
You will find if renting that apartments, apart from executive relocation apartments, are unfurnished, so if you like your couch, armchairs, coffee table, dining room table, chairs, chests of drawers, desk, swivel chair, cushions, throw blankets, etc., bring them. In order to qualify as a bone fide bedroom in Illinois (possibly the entire US) for real estate description and pricing purposes there must be a built-in closet in the room, so bedroom storage items like free standing wardrobes might not be necessary.
I would bring decorative items that you love. Books, nice vases for flowers, pictures/wall art, rugs. Bookshelves can be pricey here (imo) - bring any freestanding units. Your pots and pans, crockery, baking dishes, cake pans, roasting pan, cutlery, utility knives, etc.
Bring a good stash of your favourite tea and coffee to last you until you find something here that you like. Also any squash the DCs like. It's virtually unknown here. You can find Polish blackcurrant squash in Tony's Finer Foods in Berwyn 