The tractor won't turn over. Battery is charged, and I've even tried jumping the battery. The only audible symptom is a chattering relay or solenoid when I turn the key. I am wondering how to troubleshoot the starter/alternator. The engine turns freely by hand, not seized. Any help with an explanation of the chattering sound would be helpful.

Solenoid makes one click if its good and no clicks if its bad. Relays can chatter and if you are 101% sure your battery is putting out 13 volts then shop for a relay.

Yep, you're right. The battery measures fully charged, and my charger indicated it's charged, but when it tries to crank, the voltage drops very low. Maybe some serious internal short? Anyway, I applied 12v. From a jumper battery to the starter, and it took right off, and would run fine thereafter on the alternator output. Wouldn't charge, so it's dead again. Will replace the battery in AM, and check the charging circuit, too.
Thanks for the steer.

The ESR (Effective Series Resistance) of the battery increases and thus there is too much of a voltage drop in the battery and it cannot deliver the proper current.

Jumping a bad battery doesn't always work. In a car case, I once or twice had to remove a battery cable and jump the vehicle and then re-connect the battery.