New gun or new to you??
First check is the oddball "dog leg" firing pin.
If the strangely shaped firing pin breaks it jams up the gun and it can even be extremely difficult to disassemble the gun.
Make sure the safety is all the way forward to the FIRE position.
Check the recoil spring guide assembly to insure the guide rods aren't broken or bent.
If it's new to you look for alterations.
Stock replaced?? Check for an overly long stock screw sticking into the lower receiver.
Check the lower trigger assembly to insure it's assembled correctly.
Recoil and firing pin springs correct and not replacements??
Troubleshooting method........
Separate the receiver and the trigger group.
Remove everything from the upper receiver.
Replace the bolt and check for free and full movement in the receiver.
Replace the bolt handle, check again.
Replace the cylindrical hammer and check again.
Replace the firing pin spring and guide, check movement.
Replace double recoil spring and plate, check again.
If all that's okay, it almost has to be something in the trigger group.
Here's the owner's manual with a schematic. If you feel comfortable, disassemble the trigger group and check for broken/altered parts and proper assembly:
Thompson manual
Check book stores for the Gun Digest Books of Firearms Disassembly. These are the big paperbacks.
One of them has full disassembly of the semi-auto trigger assembly.
To start, use a piece of plastic or wood to pull down the upper leg of the trigger plate so you can slide the safety out.
The upper leg spring retains the safety and gives it the "snap".
Then wiggle and pull the trigger plate out of the trigger group.
This releases everything.
NOTE that the trigger plate tends to stick rather badly so you may need to use a nylon drift to lightly tap on the large trigger plate pins to get it out.