If the CPD bit in the CONFIG word is zero, then the EEPROM is protected and you are S.O.L.
If its set (its default state after chip erase), the EEPROM can be both read and written by any programmer + programming utility that can be configured to not do a chip erase, and only write the EEPROM memory range. Setting this up depends on your programmer, and it would be advisable to practice on a new PIC16F876 to confirm you can write to the EEPROM without disturbing the program memory or CONFIG word.
Even if the chip is fully protected, the bits in the CONFIG word can be read back, so you can simply read the chip then examine the CPD bit and the values returned for the EEPROM area to determine if its possible.