Interfacing SD card with an FPGA
Take a look at this example which uses a Xilinx FPGA. It instantiates a block which provides a SRAM like interface to the rest of your HDL logic. The actual code seems to be here (the blog is somewhat out of date)

Note that it uses the SD Card directly (as a large SPI flash, basically), without regards to any file-system. An implementation which accesses FAT(32) formatted cards will probably require an external micro-controller or soft-core processor.