I want to make a 1/4" jack a permanent feature on an old chord organ (there's tons on Facebook Marketplace). Other than buying those mikes that you use to electrify acoustic instruments, how would I, say, convert the built-in speaker into a an interface for a 1/4" jack? Would using the acoustic mike do the same as such mucking for cheaper?
You have 2 options, but both require knowledge of electronics:
the loudspeakers are driven by an amplifier. This crates a speaker-level signal, which is then split by filters to drive the various speakers. You can get a speaker to line level converter, which you can connect to the amplifier output (before the filter).
The amplifier probably has an input where the signal is at line level. If you can identify a point on the circuit board where this signal is present, you can attach your line output to this. It's possible the amplifier is on the same circuit board as the tone generating circuits, or it could be separate. I'd put a DI box between this output and your amplifier.