Nice DuMonts, so, so much for that. Don't overlook that nice RCA 630, though. I have both versions of it, myself. (630TS and 8TS30)
Looks like the power transformer has been replaced on the Cadillac set, but that's no issue here. Yes, you need a new vibrator. When Delco began to supply radios for the GM divisions (1940 was the first Delco set for Cadillac), the used a conventional vibrator, but with a freaky base. It was derived from a standard 6 pin tube base, with pins no. 2 and 5 missing. Mallory's part number was 870, but they haven't been made for years.
While I have had to quit working on other people's sets for some time now (age has pulled back on my reins) I might recommend changing the vibrator socket to a conventional 4 pin tube socket.
Look at the socket from beneath the chassis. This is the same as looking at the vibrator pins pointing upward. Pin #1 is the big one on the left. Pins #2 and #3 are the two smaller ones on the top side, and pin #4 is the right hand large one. I you replace the original socket with a regular 4 pin one, number those pins just like I just described the old vibrator base.
If you replace the socket then the pins will again number the same way as on the original socket. You always count the pins with the 2 large ones on the bottom, and count clockwise. With the new 4 pin socket in place, pin one will connect to chassis ground, and the two small pins are the leads to the power transformer. The right hand pin, (#4) looking at the pins, is unused. Try eBay for a new old stock 4 pin vibrator. It could be a reasonable source. If you get trouble, PM me, and we'll try to get you out of the grease.