Is resistor needed to drive NPN Transistor from EEPROM
However, I am unable to find the max current supplied by the EEPROM so
  I am unsure if I need to put a resistor in between the EEPROM and
  Base.




I'd assume that the current you can extract from the EEPROM is around the 400 uA level and I would also use a resistor. The resistor value would be chosen on this: -


BJT Vbe = 0.7 volts
EEPROM output voltage = 2.4 volts
Voltage across resistor  = 2.4 - 0.7 volts = 1.7 volts
Current  = 400 uA
Resistance in base  = 1.7 volts  / 400 uA = ~4k3 or above and maybe you can push it down to 2k2 to extract a little more current from the EEPROM output.


If that current isn't enough then use a darlington or MOSFET.