Hello and welcome to JustAnswer!
If you aren't hearing any noises besides a single click then you won't have to worry about the teeth on the flexplate, you'd hear a grinding noise if there were an issue there.
A single click when trying to start the engine can happen because of a bad starter, locked up engine, or lack of power to the starter motor.
If it was running fine when last parked then the engine shouldn't be locked up.
The first thing I would do is make sure full battery voltage is shown on the starter positive cable with the key in the crank position.
If that's ok then also make sure the engine is properly grounded. You can do a voltage drop test of the ground circuit using a voltmeter or you can simply use jumper cables, attach one end at the battery ground terminal and the other end to the engine.
If it still doesn't crank and positive voltage is good then the problem is going to be the starter itself. Bad contacts in the solenoid is usually what causes the problem you are having, the starter drive gear kicks out but bad contacts prevent the motor from engaging.