Seller claimed it worked and it being a nearly 100 year old vacuum do not correlate.
Verify the cord is safe and has continuity. If it is the original cord good chance the wires have broken inside or it might have a dead short, would need to be analyzed. Next thing to check would be the socket on the vacuum where the cord plugs into, it is easy for them to tarnish and lose conductivity. After that, check the on/off switch, the pins can get dirty and not make a good connection inside the switch and prevent it from turning on.
Looking at the photos, the cord looks brand new and depending on if it is an aftermarket cord said to fit the XXX, or just some random two pin cord they found somewhere, it likely is not fully seated into the socket, or the wiring to the prongs is not secure enough. Make sure you have it on the right way too, sometimes those cords were polarized and only worked one way.