Are there FPGA chips that permit to update the programmable logic from the logic itself?
If​ I understand what you're looking for, the feature is called "partial reconfiguration", and all of the major FPGA vendors support it on at least some of their chip families.

Go to your favorite vendor's website and search for that phrase.

Also, most chip families support reading and writing their own configuration PROM from user logic, so "total reconfiguration" is also an option, although this involves an interruption in operation in order to reboot the chip.