Hi, thank you for contacting JustAnswer.com. My name is Russell. I will do my best to provide the right answer to your question.
Try plugging a compatible microphone (non-condenser-type, probably) into the Mic/Line jack, and find out whether, when you record a bit of video with sound, that way, with an external microphone, it plays back with sound too.
Or we could just assume the problem is the playback speaker or the audio amplifier that feeds sound impulses to it. In which case, unfortunately, aside from trying that external microphone in it, then removing it and trying again, you don't have many options aside from having it repaired.
But if you really think it is or could be a setting or option disabling external speaker playback, then
A. look for anything - including on the screen - that looks like a speaker with a circle-and-slash over it. That means the output has been set to Mute, or muted.
B. look for any setting relating to 'volume'. Set that to above minimum at least, and make sure there are no 'Mute' controls enabled.
C. do a factory settings reset on the camera, that will restore all settings to as-released-for-sale values. (If you need instructions on how to do that, let me know.)
If all else fails, there's Sony Repair: https://eservice.sony.com/webrma/web/index.do
If the owner's manual says there isn't one, then there isn't one.
There is the Audio Level dial beneath the Auto/Man switch... make sure it is not set to 0, and set it near the middle of its range, e.g. to '5'.
There is also the Volume +/- button pair on top of the camera... try pressing the + button several times, then see if audio playback is audible.
Also try setting MULTI-SOUND to 'Stereo', in the Audio-Set menu.
I think that assuming it will need repair is correct in this case. Options, settings, and a reset of those has been tried and failed to bring the speaker to life.
Let me know how it turns out, please, at your convenience. Thanks.