I bought a used yamaha active subwoofer for my home theater, and found that one of the circuit boards inside was missing. I can't find an exact replacement for it, so I was hoping someone could either tell me how to wire it up without one or where I might find a similar style circuit board I could use as a replacement.

Two wires went from the missing board to the power switch and another two went to the amplifier.

 

Have a go at working out the circuit. It looks pretty simple. Then folks will point you in the right direction.
BUT, if you are not confident, get a qualified tech to fix this up for you as it is mains wiring and can kill you if things go wrong.

 

Could you not have a PCB made up for you? I dont think it would cost very much and would keep it somewhat original.

 

The board just looks like a convenient wire joining place. So, the board is not really needed if you just want it to go. A screw terminal block would work, with an inline fuse holder added. But, once again, if you don''t know what you are doing, don't do it! Find someone local to help you.

 

Following the circuit shown, the board is only used as a mount for the fuse and as a junction point. So yes, you can get away with using a terminal strip of a size that will fit.
But I suggest proceeding with caution because it may be that other parts of the system have also been messed with.
AND, having a PCB made for you would probably cost much more than you would pay for a new system. In addition, if you know enough to be able to include the picture you sent then you have no need of some "service professional" to do it for you.