Vinegar = An Acid, used to neutralize the battery leakage (which is a Base, in terms of pH)
An alternative is to use lemon juice - again, an acid.
Soak that under chip parts if you suspect some leakage got under it.
An old toothbrush is a good idea for scrubbing.
Sockets with green in/on them should be replaced. You can't get in the sockets well, and they will likely continue to corrode.
Badly affected chips with green are also candidates for replacement, but if not in sockets, clean as best you can, and save for later.
When done, rinse with isopropyl alcohol. This negates much of the acid and neutralization reaction, and brings the acid/base value much closer (but not exactly) to a pH of 7 (neutral). A 70% solution isopropyl alcohol is good enough, with the benefit that the alcohol evaporates, and the remaining 30% of that is pure denatured water - let that dry. Avoid tap water for final rinsing if you know it tends to have a high mineral content (but it's better than nothing if you don't have the alcohol handy). Denatured water from the store is also an option (exact neutral pH of 7, no minerals).
Check also that any through-hole solder point (a via) conducts when it transfers the signal to the other side of the PCB.