You can use the photo machine with regular office paper but it doesn’t crimp as nice on the back. Worst case you could double up the paper you’re using.
But what you can also do is print photos on thinner paper stocks. If using a laser printer, just choose any paper that is 32lbs or less. If using inkjet you can print photo quality on “presentation paper” — which is a much better option for making buttons than buying a photo machine. Because presentation inkjet paper is less expensive, and much easier to work with.
Photo machines are made specifically for photographers making on-site photo buttons from event printers, and have to use very thick stock.