A quick update, I tried an amalgam of what Elfen and Uniserver were suggesting since I don't have all materials readily on hand. I cleaned the board all over with CRC QD electronics cleaner, waited for it to dry, then cleaned the areas around the sound chip, audio jack, and speaker connectors with 91% isopropyl alcohol and a spare toothbrush. Once that had all dried off, I used Pledge (I checked the ingredients and it does contain silicone oil) front and back on the motherboard and then put it on a pizza pan in the oven for 15 minutes at 220 degrees Fahrenheit. Turned it off and opened the door, then 15 minutes after that put it back into the case and plugged everything back in.
Now it doesn't appear as though there's any sound at all. The speaker no longer pops, it just make no sound whatsoever. I checked the audio jack as well with a pair of earbuds, although it makes some crackling when I put it in or out, no actual sound from the system can be heard there either. It looks as if the whole computer has gone mute.
I'm not sure what to check next. The power supply seems solid and the fact that both the speaker and the audio jack don't work point to something within whatever controls the LC III's sound in a general sense, either the audio chip or some trace coming off of it. I'm not familiar with PCBs and don't really know how to troubleshoot if it is a broken trace or something else (I'm not even strictly 100% sure of what the audio system consists of, to be honest) but if anyone has any suggestions I will try to carry them out as best as I can.