The pick up on my son's Martin LXME won't work

That serial number would correspond to a guitar built in 1959. The model will be either be stamped on the neck block seen from inside the guitar or on the label visible through the sound hole. Martin's phone number is 610-759-2837 and they may be able to provide more specific info.

Your guitar was made in 1973.

Made in 1956.Value will depend on model and condition. Your area of the country will affect the value as well as local markets vary a lot.

if you look inside the cavity you will see a black nylon insert that is velcrod to the base. simply pull it off its velcro , swap the battery out , and then revelcro it. Super simple.

your pressumption would be correct. The Thinline 332 runs under the bridge saddle, and the Piezo is attached ti the inside of the guitar body, tilt your guitar and have a look inside, you may be able to see it.

the only other solution would be an electrical short in the wires running from the tuning box to the input cable hole. take the tunner all the way out and make sure there are no loose wires. Make sure nothing is preventing the wires from making good contact.

Talk to your sound man... It is possible he has a compressor on the line. A compressor adjusts the volume dynamically and delays tweeking and releasing as a setting. Your guitar should NOT run through a compressor.

The little latch is above the word "Fishman" right in the center top of the tuner. With a fingernail, pull the latch toward the center of the tuner and the whole thing pivots out exposing the batteries.

First, make sure the volume is up on both the guitar and the amp. Obvious, i know, but it wouldn't be the first time a guitar's volume was overlooked.
Second, is there a crackle or a pop or anything if you plug the guitar in to a live amp (one that's already on and with some volume)? If not, then it's likely the cord or the amp, not the guitar.
Even if there is a sound when plugging in, try the guitar with a different cable (one you know works) and a different amp, if one's available. If there's still a problem, then it's probably the guitar.
Standard troubleshooting procedure is now officially out of the way.
Okay, so, I'm not terribly familiar with all the sigma models--in fact, I'd imagine it's hard to find someone who is--so I'm probably going to need a little more info.
Firstly, the model would probably help, and might answer the rest of these questions. Since Sigma wasn't always so helpful with serial numbers and such, we probably won't be able to get an exact ID on the guitar, but we might get a ballpark.
Next: Do you know if it's a passive or an active pickup? Is there a little light and/or button somewhere on the electronics that might be a battery indicator? If so, and nothing lights up when you push the button, see if you can change the battery easily. Sometimes it's obvious how to do this, and sometimes it's not. If it's not obvious, see if you can post a pic and we'll work it out.
If the battery is still good, then it's most likely a problem with the wiring connecting your pickup(s) and/or microphone(s) to the guitar's preamp electronics. With acoustic-electric guitars especially, it's hard for the actual pickups or mics to go bad without something happening to break them (like being dropped or whatever), and it's much more common that a wire gets pulled out or disconnected. Still, things do get old, and they do break.
See if you can tell what kind of pickup(s) and/or mic(s) you have in the guitar and take a look for any hanging wires. You may get lucky and just have something to re-attach, or it may be something easy to replace. Or, you may be stuck trying to order replacement parts from Martin.
If none of this finds the issue, and we can't figure it out with pics or model specs, I would recommend taking it in to a local music shop and having them work it out with Martin for you--they won't gouge you on the price, and they may even explain how to do the repair yourself.
Let me know when you have some more details or if this worked for you. Good luck!
(P.S. I completely feel your pain; my Ovation 1985 collector's series was recently without electrics until I tracked down a replacement pickup and rewired it.)

REMOVE THE STRINGS FIRST