What kind of sensor am I looking for?
I think your best bet is piezo sensor to which a peg is attached. Unlike mechanical switch it would not change "bouncing" property of the peg much. And it should not have much cross-talk with other pegs if amplification gain is right.
You can drill a hole in non-sensing part of the sensor and mount a bolt with a nut. Or, even better, solder a piece of brass tube to it, which would give you a conduit for LED wire (the second is the tube itself).
Interfacing the piezo sensor is simple and it will give you a pulse than can be stretched to light up the LED.
Here is alternative idea that might be somewhat cheaper to implement. For this you need long bolts mounted on a back panel and coming forward through slightly larger holes allowing for a tiny bit of bending. You glue strong magnets to the bolts right behind front panel and put coils around them.
From here the idea is clear - you amplify the coil current and set a threshold for strike detection.
In either case I must agree with @tony-stewart that giving the amount of pegs in the typical game it won't be cheap no matter which way you go.
Here's another one :)
There are digital accelerometer modules available under a buck. You'd need the chips with lowest possible range (something like 0.5g). Use same setup as above but glue these modules to the pegs instead of magnets. You don't need to read them, just program "interrupt on tap" feature. This will cover all the analog circuitry required in other solutions and you'll get clean digital signal right away.