Can FPGAs dynamically modify their logic?
Yes, I know that at least Xilinx has parts that support dynamic reconfiguration, and the other major vendors probably do, too.

It's a major undertaking to do it, though, so you really need to make sure you need it. You need to partition the chip physically into two or more areas, at least one of which is non-reconfigurable, and physically "pin down" all of the internal interfaces among the areas so that the synthesis tools can make all the right connections.