The Thompson would have had to fire .45ACP blanks. I think it did have to eject them, but it's possible the shutter of the movie camera might have obscured them. If you've seen westerns when stagecoaches go by, and the wheels appear to be either frozen, or turning in reverse, you have noted the effect a 24 f.p.s. motion picture camera can have on some moving items.
There is a coffee table book on COMBAT! written by Jo Davidsmeyer in which the author relates a story almost identical to bannockburn's post. Rick Jason was offered the Thompson, but he was a hunter & reloader and knew what it was like to carry a gun around for a long period of time, and he deliberatly asked the prop man for the lightest gun they had, which is how he wound up with the carbine. All the other guns worked OKAY with the barrels blocked for blank use, for some unknown reason the Thompson was, as Vic Morrow said, a "jammomatic." That is not why it was used in short bursts; it was normal to fire the Thompson in short bursts due to what's called "walk up." A submachinegun, when fired by a right handed user, will tend to have the barrel go up and right due to collective recoil. Users were trained to hold the weapon properly and to keep bursts with SMG's down. This also helped conserve ammo to an extent, but the real reason was to keep barrel rise under control & fast reacquisition of target.
Having said that, I have seen a few scenes in COMBAT! where the Tommygun did jam, and the solution was a quick cut to a different viewpoint.
I recall Saunders carrying a Garand in the D-Day landing episode titled "A Day in June" and a M-1 Carbine in the episode "The Squad," but I don't recall any other episodes where he carried something other than the 1928A1 Thompson.
The prop Tommygun was made out of wood by the MGM prop shop, and it was made up because Morrow complained about the weight of the gun over the period of a long work day.