The construction of a vinegar battery is quite simple and easy. The materials you will need are a zinc strip. A galvanized nail works great. Some copper wire and a plastic container with a top of some sort, vinegar and a piece of copper if you have it. If not just strip a bunch of the copper wire and use it.
The copper will serve as the positive terminal of a battery and the zinc/galvanized nail will serve as the negative terminal of a battery.
According to my multimeter, a vinegar battery will producing about .8 volts. Keep in mind the amps available will be very small. This isn't a substitute for a cranking battery even if wired up in series to over 12 volts.
These batteries are only suitable for running leds and super small electronics like a calculator etc. And even for these small task you will need to wire up multiple cells in series and parallel.