My 1909 Buick Century starter makes a clunk sound, when I turn the key but wont turn over, could this be the solenoid?

The Starter motor can keep running because of a number of things. It receives its power from the large contacts (switch) in the starter solenoid. WHen engaged, this starter sol. allows high current 12 volts to run the starter motor.
A number of things can cause the starter solenoid to stay engaged:
A) stuck starter solenoid mechanicals. probably not the cause
B) more likely a defective start position on the ignition switch in the ignition switch
C) possibly a gummed up ignition key tumbler
I would start troubleshooting with a schematic. Make your own or purchase a service manual. IF you pull off the wire from the switch (in the center console) which actuates the starter solenoid, and the starter motor continues to run, then its not the switch or the key tumbler. its the solenoid.
Troubleshooting is the logical process of elimination.

Could be a short in the key switch, the wiring between the switch and the starter, or the starter solenoid. If you disconnect the small wire on the solenoid and the problem continues, its probably the solenoid.

The solenoid located on the starter motor is going bad. You do NOT have to get a whole starter to fix this. Just replace the solenoid. You can get that part at any auto parts store.

Firstly it is advisable to energise the starter solenoid manually. If the engine cranks the fault is - immobiliser, ignition switch, relay, supply, wiring or connections.

If the engine doesn't crank the fault is likely the starter motor or solenoid.

If you battery is charged, sound like a starter problem or a ignition key switch problem.

Check if the starter solenoid is faulty, check the power to the solenoid and if there is no trigger to the unit then the key switch contact can be checked. You can cross a +12 to the solenoid, if it wont work then the unit is faulty. If the solenoid works but does not power the starter motor then the mtor brush or commutator can be worn out. You will then need to remove the starter and overhaul.

Check for power at starter solenoid on switch side (purple wire) should have 12-12.6 volts if you have power to the solenoid and it won't do anything then the starter is bad.

If you get power to the small wire on the solenoid, when you turn the key to the starting position, the ignition switch is ok and you may have a bad solenoid and/or starter. Most parts stores can test your starter. Also check the "crank" fuse and ignition fuses.

have battery tested first, if you have a good battery, replacing the battery is a fairly simple task with good jack, jack stands, and a few metric sockets and wrenches.

this sounds like the transmission locking torque converter clutch is not disengaging, this will kill the engine. This problem is most often caused by a defective TCC solenoid.