EEPROM Reading/Writing without Use of Programmer
Whoever told you that the EEPROM had to be programmed off-line using a separate programmer is not correct. That is just one way of doing it but it is not the only way. This person may have been thinking of older technology such as EPROM memory which is often easier to reprogram out of circuit due to the special voltages or timing pulses needed. Yet even EPROM can be done right in circuit.

For your application it may be actually easiest to utilize a CMOS static RAM chip with parallel bus interface and a small coil cell backup battery to provide the non-volatile storage for the password. I say this because it would likely take less logic to support such a setup. 

On the other hand there are various types of serial interfaced EEPROM memories that operate off a single voltage rail and can be programmed and erased right in circuit. The two most common interface for these are SPI and I2C. You will find that for a descrete logic design that the SPI type will be easier to operate. Look for a part like a 93C46.

Serial EEPROM type devices are convenient but do require a long pulse time to support the write operation on each data item. For this reason you may want to look at a serial FRAM type device from Cypress that has equal read and write times. 

With all this talk of using non-volatile memory devices for your project there may very well be a much more appropriate approach. One that requires a lot less logic circuitry!! If you keep your password design simple such as limiting to three digits (0 to 9 values) you could easily store the password into three 4-bit registers. These registers could be wired up into your logic circuit in a way to have a backup battery power source keep the registers powered and thus preserve their stored content whilst the main power to the door lock was absent. I think with this approach you can still learn a whole lot but at the same time show how engineering solutions can often be way simpler than the way that pops into mind when we have the noise of the modern methods poking away at the back of our brains.