It'll most likely only be one of the ICs, or the PIA. What you want is a 4049. Great plains electronics should have them, PIAs, sockets for both, or any regular electronics site like digikey will have 4049s as well

I'm not completely sure of the difference myself, though I'd lean towards the former as it doesn't have any letters in between the numbers. the HC letters mean something about the speed and power of the data lines. There's also HCT, LS, F, etc, but basically you need to compare the datasheets and know what the numbers mean. I'd just email GPE and ask what you need for this application, he should be able to tell you

It's all going to be 5v logic on this board. Most chips operate at a single voltage level, especially logic chips, so just look at what voltage is powering the chip

The 4049 reads the signal coming back from the matrix, so without the connectors you're just going to get a high on all the inputs and a low on all the outputs (its an inverter so).

Each line goes only from a PIA pin to a pin of the 4049.

Looking at the schematics, pin 7 of IC15 (the lower 4049 I assume) goes to 2J3-8, which it says is row 2, not 7. Unless you're just counting the other way?

The output of the 4049 should always be the opposite of the input. The pull up resistors (eg R78) default the input to high, meaning the output is low until a low input comes in via the jumper. What are you jumpering from when you jumper the row inputs? The column? ground?

PIA pin 3 is read by the CPU to tell whether the switch in that row is closed or not. If you're sure you've got your input pins right, and you can see the signal going into the PIA moving, but the row isn't reading, then the PIA is most likely bad.

Is the PIA is socketed?

Is the leg of the chip perhaps broken off or the socket not soldered correctly? All those things should be connected together and showing the same signal, so if they're not there must be a break somewhere

Not really much risk. Try resoldering the socket. Or swapping the pia with another.

Done well, you wouldn't see those darker brown spots on the pads, just the nice silvery color. I can't tell from the picture whether that's copper (under the silver) or something else, but if it's copper that's still okay, just might be a bit harder to solder to.

On the back it looks like two pads have fallen off. That probably means more are damaged. There's a little metal ring that goes between the pads on the front and back. If that's missing then a trace on the front won't be connected to the trace on the back. Need to make sure to put enough solder on that it reaches both sides.

What pin #s are those two that have lost their pad?

So it looks like 14 is supposed to go to IC18, pin 1 (like that jumper does) and 15 is supposed to go to IC18, pin 5? It looks like the traces for both of these are actually on the top of the board, so if the trace itself is fine you should be able to just solder them on the top and not need the jumpers. Check continuity from the pads on the top to those pins on IC18 and see if the traces are fine. Since two traces are already bad, there's a good chance more are, including the problem you're having. I'd recommend when installing the new socket to make sure to try to flow the solder through the board so it goes on both sides of the pad, and check continuity from every single pin of the socket to the part it's connected to on the board, resoldering or adding jumpers as necessary. It may help to use SIP sockets instead of a standard DIP socket as they're easier to see the solder on the top side

I'd replace that PIA to be safe, or at least have a spare on hand.

I sadly don't really know of a good source. I have a big stack from dead boards I've attempted to repair to tide me, and usually it grows faster than I need them :/