The system they need is to let me bring up the TM30 that the juristic office submitted for me. Getting the juristic office to provide a TM30 receipt in a timely manner was (for me at least) a bit painful.
Up to the airline in the country you depart with an end destination of Thailand. On my last trip here, I flew Singapore Airlines to enter on a visa exemption (US), and they required a ticket leaving Thailand within 30 days before I could board.
I did this a week or two ago. I recommend making an appointment online. You’ll need printed hardcopies of everything (passport, visa entry stamp, TM30 receipt, appointment confirmation email) in addition to your passport. You won’t get any of the hardcopies back, so make extras if you want them. You can download the TM7 form and fill it out, but there’s a second form you’ll need to fill out also - not sure of the form number, but it’s quick and it and the TM7 are available at Laksi.