How to 3D print an ID card
FDM printer?
If you want to print one, maybe you should outsource it (let it print the tag on both sides), even the most affordable printers are in the \$100 - \$150 price range. If you want a printer and use it also to create ID tags, you could go for an FDM printer. Considering your request of having the tag inside (and through) the ID-card you need a dual filament option (one or two nozzle arrangement). If the tag can sit on top you can print it with a filament change with a single filament single nozzle printer. But, don't expect to get crystal clear prints (see experience printing signs below)!

Alternatives
As an alternative, you could print a blank PLA ID-card and laser mark the tag onto both sides, see e.g. this video.

If it is a small batch you can also consider printing/lasering stickers and stick these onto blank ID cards.



From experience
I've done some signs with black letters on a white background for "on-lay", inlay and through arrangements using a more expensive (for home use) dual extruder 3D printer (Ultimaker 3 Extended about \$5000,-) with PETG, but the results were not very satisfying. Usually the black smears out on or in the white no matter tweaking the options. Considering the size of an ID-card, the amount of tag squares, this is even more likely to happen when you print at that small size (the signs I printed were sized similar to the "A5" paper standard). 

From my experience I would say that a 3D printer may not be the best solution for your task.