I've not seen documentation that negates this option, but wasa wondering if the 5V mini (8MHz) Mini could run a SD card reader and card...
Yes, you need to get an sd card shield or attachment. Make sure the hardware has a 3.3V power supply on it for the SD card and level shifter to safely make 3.3V SD card talk with 5V arduino. Adafruit has such a shield for sale.
There is a 3V3 8MHz Pro Mini board which wouldn't need a level shifter (or a separate 3V3 supply).