If you hear a click, that could be the battery starter solenoid or the fuel shut solenoid pulling in.
I'd start by jumping the starter to see if it spins the motor. There should be a small pin on it that you apply +12v to and it should make it crank. If it just clicks, it is either a bad battery or a bad starter.
The first thing I would check would be the battery connections for corrosion, that can act like a resistor that won't allow amperage through, but will show full voltage. Corrosion is usually at the battery, but can migrate through the cables if left long enough.
Oxidation and rust can occur at the starter and ground ends, so it never hurts to disassemble and make them clean and bright if there is any doubt.
Starter solenoids usually give plenty of warning that they are getting marginal by requiring a few attempts to get them to engage, by the clicking.
The battery could have a bad cell, that would allow it to show full voltage, and a charger on it would quickly indicate a full charge, but any load would immediately drop it to about 9v; I have a pickup battery that's doing that now, need to swap it out.
A few quick checks with a DMM (and I consider a DMM an essential bit of kit for
anyone who knows which end of a screwdriver to use, especially since Harbor Freight gives them away) should indicate which way to go.
