JT: If you follow the cable from the negative post of the battery, I think you will find it connected to the tractor's frame: the frame is the ground reference for just about any vehicle. If your voltage regulator does not have a terminal for a ground wire, then it receives its reference from the connection of the voltage regulator's case to the frame/body of the tractor. You should clean all connections from the V-R case to "ground." If your V-R is mounted above the engine on the nose of the tractor, and you lack a terminal for a ground wire, I should think the ground connections include where the nose bolts to the frame of the tractor.
If you haven't already, I would clean the connections at the solenoid and the negative terminal of the battery at the frame. Again, if you haven't already, I would make sure your battery's posts and clamp terminals are clean and tight.
Note: If your solenoid is clicking, this means that your key is working and that you are getting 12 volts to the "magnetic switch" as the solenoid is termed at the small wire and that the ground at the solenoid is good; but if the starter isn't turning over then either the big wire at the solenoid isn't being switched, or more likely, the connections (especially at the "grounds") aren't secure enough to carry the higher current loads on the heavy gauge wire.
There is, of course, a third possibility, namely, that the starter/generator is bad, or there is a problem with the wiring to it which runs through the voltage regulator.
It is difficult to solve electrical problems by mechanical means, even when the root cause is mechanical. You need to use electrical tools to properly diagnose electrical problems. So if you have a volt meter, try to get readings as follows:
With the volt meter set to the range of 0-15 or 0-20 or 0-30 or 0-50 volts DC (whatever is the closest range for your meter) perform the following tests with the battery hooked up in the tractor. The first three tests are preliminary diagnostics, you can start at step 5 if you want to.
1. Take two readings: Check across battery terminals you should have a good 12 volts to start. Check at the posts, then check again at the battery cables. If the readings don't match, clean the battery posts.
2. Take a third reading: Check from the frame to the positive battery post. Find a spot you don't mind scratching the rust from in order to make good contact to ground with the black, negative lead of volt meter; you should get a reading that closely matches the first two readings above. If the reading is significantly lower, clean the connection where the negative battery cable connects to the frame.
3. Take a reading: If you're curious about how well the voltage regulator is grounded, then take a reading from the ground terminal or V-R case to the positive side of the battery, again, you should register close to 12 volts. If you don't have that reading, then clean the connections necessary to achieve it.
4. Try to start the tractor.
5. If the tractor doesn't start, and you're getting a click at the solenoid, take two readings across the BIG wires at the solenoid. This step may require an assistant. First reading: without touching the ignition key, read the voltage from the positive terminal of the battery to ground (as in step 1 above) you should have 12 volts. Second reading: while holding the volt meter terminals to the positive terminal of the battery and ground, turn the key --the voltage should drop significantly. If the voltage at the positive terminal of the battery does not drop, then there is no current being drawn by the starter/generator, that is, it isn't even TRYING to start and either the solenoid switch contacts are corroded beyond usefulness, you've lost your ground, there is a problem with the starter/generator, or with the starter/generator circuit (which includes the voltage regulator). This same test can be duplicated if you have lights and they work. Turn the lights on and try to start the tractor, the lights should dim. If the lights go out, the tractor has lost its ground, if the lights don't dim, there is a problem with the starting circuit-- the solenoid, wiring, V-R, or starter/generator.
6. If you're curious about the condition of the solenoid, take a reading: move the red, positive lead of the volt meter from the positive post of the battery and move it to the terminal on the solenoid that runs to the battery. Whatever reading you have at the positive post should be replicated on the downstream side of the "magnetic switch." Of course, if the voltage reading isn't dropping as it should, this test is inconclusive; a better test is to disconnect both "big wires" and apply 12 volts to the terminal for the little wire. Switch the volt meter to reading ohms and take two readings: first reading without applying power to the IGN terminal should be infinity; second reading, with 12 volts applied at the small-wire IGN terminal (the solenoid's frame must be grounded) should be continuity (0 ohms) or close to it. If you are reading continuity across the big terminals with the switch activated and disconnected then your solenoid is working.
If you work through all your grounds, and the solenoid checks out, then the problem is either at the voltage regulator, the starter/generator, or the chassis wiring itself. If you're not comfortable working on electrical stuff, I would probably seek professional help at this point. Report any results and we'll try to help you.