First, welcome to OSW!
For an ID badge, the two that come to mind (having worn both at work) are either a plain badge lanyard, where the ID card would hang at sternum height, or a little retractable cord thing that normally clips to your belt.
Hope this gives you some ideas!
"Women uh... women sense my power and they seek the life essence." General Jack D. Ripper
You might consider a small straight pin- used in sewing. If you don't mind piercing the ID card, you could poke the pin through- to get an idea of where you'd need to bend it. Then, take the pin out and bend it just below the head. Then poke it back through and with the point of the pin at the back, pointing down, stab it through the lapel. This way you could take it off whenever you want and the hole should be tiny- if any remains- in the fabric of the suit/shirt.
The only thing that comes to my mind is the DML Molle Clips from the M16 pouches and the other Molle clip from the M9 bayonet. But how can you do it with it because not all suits has working pockets on left chest.
The top of your I.D. has the detailed rectangular supposedly the provision for the I.D. which in 1/1 is where you loop the piece of plastic, fasten it then the end has the mounting clip for the clothes. You can fabricate a piece of long plastic trim to size, fold and use a pin pierced on the plastic and attach to the suit.
on the super-quick-and-cheap, a heavy duty staple will work. one side of the fold goes through the card and the other can be used to attach the id to a pocket, etc., and it requires no glue, etc.
The garter belts sets have a clip that would work perfectly. Its generally oversize for a garter but would be perfect for an ID clip