Yes, but part of it will be specific to every game. It involves disassembling the game's code so you can find where to add some custom code into match attempt (for example), as well as the location in RAM of the score and initials.
If you can locate or add one single output pin that the CPU can control, hardware-wise that should be enough. Connect it to a 5V RS232 or USB transceiver. For the software, you need to either find unused ROM, or expand the available ROM. Probably don't need much more than 100 bytes or so for the code additions. Hack code into the game that reads the highscore / initials out of RAM, formats it into ascii text, then send it off to any kind of terminal program on your PC at 19.2kbps or whatever speed. I could point to some code examples for this part on a 6502, but it seems 68xx is more commonly used on pinball machines.