PICkit3 MCLR to Vdd Pull-Up Resistor
No the pullup resistor isn't required, however without it, unless there is something else to perform its function, the chip won't run under normal operation.

The purpose of the resistor is to ensure that MCLR is held high so that the chip isn't in a RESET state, which allows it to run.  If you don't have that resistor you would need some other system to ensure that it is held high - maybe a reset / power management chip.

The fact that you can't program won't be related to that resistor, instead you should look to other connection or circuit issues, such as a lack of decoupling capacitors, not all the power pins connected, or other components on the programming pins interfering with the communication.