"<font color="#6699ff">[b:654c4848f0][i:654c4848f0](Original wiring on tractor)[/i:654c4848f0][/b:654c4848f0]</font>"
Take a look at the [b:654c4848f0]Electrical Wiring[/b:654c4848f0] diagram below.
Note that the electrical system was originally a POSITIVE ground system.
Note the braided battery cable is connected to the POSITIVE battery post and grounded to the battery box.
Note the solid battery cable is connected to the NEGATIVE battery post and to the solenoid on the cranking motor (starter).
"<font color="#6699ff">[b:654c4848f0][i:654c4848f0]If you look on the seat rim, there's a broken off wire. Maybe that was the ground?[/i:654c4848f0][/b:654c4848f0]</font>"
Note the only small black wire in the battery box is connected to the NEGATIVE battery post and to the SWITCH, STARTER SWITCH, and ELECTRICAL OUTLET SOCKET.
"<font color="#6699ff">[b:654c4848f0][i:654c4848f0]Could I just put a new wire in that and then connect it to the negative cable clip where you see another wire, probably the same one?[/i:654c4848f0][/b:654c4848f0]</font>"
Yes, if the new wire is connected to the solid battery cable that is connected to the NEGATIVE battery post.
"<font color="#6699ff">[b:654c4848f0][i:654c4848f0]Would the tractor have a hard time starting if a bad ground?[/i:654c4848f0][/b:654c4848f0]</font>"
Yes.
Instead of using the braided battery cable grounded to the battery box, you might consider using a solid battery cable grounded to the rock shaft.
"<font color="#6699ff">[b:654c4848f0][i:654c4848f0]I'm getting nothing when pushing the starting button.[/i:654c4848f0][/b:654c4848f0]</font>"
Take a look at the [b:654c4848f0]Electrical Wiring[/b:654c4848f0] diagram below.
You might consider replacing the solid battery cable with one that has a better connection to the NEGATIVE battery post.
Using a test light, check for power at the solenoid terminal connected to the solid battery cable connected to the NEGATIVE battery post.
Shift the transmission into NEUTRAL and lock both brakes.
If there is power to the solenoid terminal, then "test" the solenoid.
If the solenoid engages the starter, then move on; otherwise check connections and repair/replace the solenoid.
Check for power at the STARTER SWITCH terminal connected to the black wire from the solenoid.
If there is power to the STARTER SWITCH, then move on; otherwise check connections and repair/replace the black wire.
Push the starter button and check for power at the other two terminals with black wires connected to the SWITCH and RESISTER.
If there is power at both terminals, then move on; otherwise check connections and repair/replace black wire(s).
Hope this helps.