You don't insert the card in the machines.

You tap the card like you would newer debit cards and Point of Sale (POS) machines.

I'd look into whether your apartment building could add that as an option, or if not, look into procuring the card that works with the laundry machine manufacturer and see if you can preload it somehow yourself.

If the manufacturer doesn't make those cards, you might try cards from other manufacturers to see if those work.