The PSU is connected to a normally
closed switch? That seems like a very odd setup.
The system needs to ensure it'll only work, when the authorised user is at the control box.
Well... a great way to do that is to make the user push
and hold a button (or some sort of dead man's switch, anyway). In which case, you'd be better off with one-shot access control such that the access token gets immediately returned to its resting position (e.g. the user's pocket, or can be on a lanyard that never leaves contact with the user). Also, you can get COTS token authenticators for dirt cheap. (You might even consider using a fingerprint reader, as Nominal Animal suggested.)
You'll need two relays, though. One can be as dirt-cheap of a 12V SPST as you like, the other needs to be able to control your LED but can also be SPST. (SPDT is fine, you just won't use the NC contact.) And the momentary switch does need to be dual pole (DPST or DPDT).
Hook it up like
this.
The "black box" (with "in" and "out") is the control box of a generic, COTS Access Control System (ACS). I checked this time, and you
should easily be able to find one that has both NC and NO terminals, i.e. one that momentarily
interrupts voltage and one that momentarily
supplies voltage. It will also give you the 12V from mains that you need for the relay logic. The first/upper momentary switch is the token reader. The effect of this in real use is that "out" goes briefly to +12V when an access token is used, with a delay (usually a few seconds) that can be adjusted by a pot on the controller. The lower pair of switches are linked, though this isn't visible in the schematic; they're the dual pole momentary switch used to turn on the LED.
The way this works, when the ACS is activated by a valid token, voltage is briefly supplied to the first relay's coil. The momentary switch is connected in such a way that pressing the switch will "latch" the relay by connecting the relay's NO back through the coil. Thus, if the ACS is supplying power, holding the momentary switch will
keep power connected as long as the switch is held. The relay also supplies power to the second pole of the switch, which in turn drives the second relay. This means that, if the first relay is open, pressing the momentary switch does nothing... and the momentary switch can't close the first relay, it can only
keep it closed. The only way to close the first relay (and thus, to turn on the LED) is for the ACS to supply the initial voltage pulse as when presenting an access token.
Thus, the LED can only be turned on if an authorization token is presented, and while a user holds down the momentary switch. The moment the switch is released (unless still within the few seconds of authorization), the system requires the token to be re-presented to turn the LED back on.
Now... you
could build this with the part you linked... the very
expensive part that requires the token be left in place during operation... or you can use a COTS RFID ACS that costs a fraction of the price. 🤷
Of course, this assumes that you need the LED turned on for however long the user presses the button. If you're okay with it turning on for a fixed time interval, you might (depending on what interval you need) be able to dispense with everything but the ACS and second relay. And you might be able to use a timed relay if you can't get a suitable interval out of just the ACS.
Oh, and, if you're feeling clever, you might go ahead and get an ACS package that comes with an electromagnetic door lock and hook that up in parallel with the second relay coil, such that the room with the LED can't be entered while the LED is turned on. 🙂 That might also allow using a fixed-on interval even if you otherwise couldn't.