There are at least two separate potential issues here, copyright and trademark.
Copyright
Unless a logo is so simple that it cannot be the subject of copyright protection (the Target logo is a well-known example) any logo will be protected by copyright, unless perhaps it is old enough that any copyright has expired. (The classic coca-cola red script logo would be an example of a logo whose copyright has expired.)
Reproducing a copyrighted logo without permission would be copyright infringement, unless an exception to copyright applies. In the US this would mean fair use. As this would be using the whole of the logo, would not be transformative, and might impede a market that the copyright owner could choose to exploit, it seems unlikely to qualify as fair use. Those non-US exceptions that I know of, such as fair dealing, also do not seem likely to apply. It seems likely that if a copyright owner brought a copyright infringement suit against such a business, it would be likely to win.
Of course one could attempt to obtain permission. I have no idea how likely the various copyright owners would be to grant such permission.
Trademark
Trademarks are protected against use "in trade" or in "commerce" without permission. The use of a trademark which soemoen else owns to identify your product, to pass off your product as actually made by the trademark owner, or to falsely imply or suggest that your product has been authorized, approved, or sponsored by the trademark owner. An ad for "Custom {brand} ID badge with authentic {brand} logo" might well be trademark infringement.
A well-worded disclaimer making it clear that this is an independent use of the mark, in reference to the brand, but not by or authorized by the brand, can sometimes avoid trademark infringement liability, dependign on the exact facts.
Making a custom badge to order, without using the mark to advertise or label your product, would probably not itself be trademark infringement.
Accessory to Fraud
If you make a custom badge for a person who is not in fact a legitimate employee or agent o the company whose logo is used, might well cause problems. If that person used the badge to falsely pass himself or herself off as an employee, that person might have committed fraud or some other offense. And in that case you might be considered an accomplice, particularly if you have made no attempt to verify that the person was in fact an employee or agent of the firm whose logo was used.
Some verification policy might be a wise idea.
Conclusion
There are some legal risks to such a business, but trademark infringement is perhaps the least of them.
