Just about any that can handle serial data and has at least a couple of hundred bytes of RAM (not actually essential, but it makes it simpler) - a PIC, arduino, raspberry pi, just about anything...
A typical RFID reader interface just sends a stream of serial data; either conventional serial, or wiegand format - two data wires, pulsed for one and zero bits respectively.
You wait for a full card number to be received, then compare it to the list of valid cards stored in the device memory.
Or for a card that contains "credits" you need a Mifare style card and reader/writer.
That gives you the card number, but also allows data to be read and written securely, you you can check a credit count and write back one less to "pay" for a service.
This is an example ready built Mifare reader/writer with a serial interface:
NFC reader/writer with RS-232 serial port. It can read and encode NFC Tags type ISO14443 A and B, MIFARE, Felica, NFC Forum types 1-4. This product is out of production. Alternatively, we recommend the NFC reader ACR1281S-C1.
Or you can get a module such as this, with a matching antenna, and build your own device:
Look at the data sheet link in the module listing for the technical info.