You don't store the code in the FPGA, you program it into the configuration chip memory. The FPGA loses its configuration when powered off and is reconfigured at power on by the configuration chip.
So, it's the configuration chip that you want, not the FPGA. Well, you also need another FPGA to go with it of course.
Configuration chips are small and cheap, and can be programmed in-situ (probably with the same tools you have with the dev board, or even byhacking a connection to it) so you really don't need to go pulling them off dev boards.