I stumbled off of a step ladder and crushed this ductwork. It's in a bathroom and runs along the floor on a concrete slab between joists. The house is 50 yrs old. The ductwork looks like it's galvanized. How do I repair or replace it? The end of the run continues under a wall. Is there a seam on the bottom? If so, I probably ruptured it. 
How to Fix Ductwork that I Crushed
ducts
Related Solutions
If you can't get to both sides, you could drill a 1/2" hole (which may be tricky) and then use a knock out plug that is designed to snap in from one side.
You could then use something like the strain relief described in gregmac's answer to mount the thermo couple on the plug and then snap it in and seal the edge with foil tape.
Chances are the end of your horizontal stack into the box has a lot of tabs that are holding it together. I would leave this in tact. I would remove the duct tape (that shouldn't be used for ducts) and release the two pieces of horizontal duct that are being held together with said tape.
These are probably just smashed into each other and chances are you won't even have to have tools (maybe a screwdriver if they put a couple of small screws in) to remove this. Once you break these two pieces apart you can add insulation on your wall.
Note about the insulation - Please please focus on the lack of insulation I see in your joists above the cement going all the way around. This is way more important that insulating the walls of your basement.
Once you have insulated you may need some tin snips to cut back an inch or two on one of the horizontal ducts. You also may be able to smash the duct into each other sufficiently - this depends on how deep your insulation is. Also if you don't want to deal with cutting metal most big boxes will cut a sheet before they sell it to you. Not sure which one I would take them but possibly the one on the right. But anyway these are dramatically easier to cut when unrolled.
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Best Answer
Looks like common 3 1/4" x 14" rectangular tin vent pipe, you should be able to buy it at any good plumbing shop. It sometimes comes with a locking seam along one edge but HVAC professionals often buy non-seamed pieces of various lengths.
You may be able to beat that back into shape and use listed duct tape to patch it up. If not you will need to find the next upstream connection and replace the whole length.