This is the large chain sprocket on my snow blower. Different brand than yours, but should be similar. You can see the quarter inch square key that fits in slot cut in impeller shaft and slot cut in sprocket hub. Then the set screw tightens down on it. My key is 1/4" X 1/4".
Mine also has a 1/4" bolt through the hub and the impeller shaft. Yours may not have that.
If your key is missing, it will need to be replaced.
I bought a 12" piece of 1/4" X 1/4" stock at local hardware store and cut a piece to length as I need one. I usually cut a piece to length, then chamfer (Bevel) all 4 edges on one end of it to make it a bit pointed.
I align the two slots, then push the new key in. Usually have to tap it in with light taps with a fiber glass or plastic hammer so as to not damage the end I am pounding on. Might need a temporary spacer or a pry bar behind the sprocket to keep it from being pushed too far back. Align the sprockets and chain and tighten the set screw.
Turn the snow blower impeller by hand a few revolutions to see if all is aligned as it should be.
If your key is still in place, then check to be sure sprockets are aligned, and set screw is tight.
If your chain is #40 chain, I would buy a 10 foot roll of new number 40 chain and cut out a new chain to replace the old one. I think I paid $22.00 for the last roll I bought. Nice to have some new chain on hand on a snowy Sunday afternoon when a chain breaks.
I do not have a chain break so I figure out the number of links I need, then grind off the rivet head on one side at that point and drive out the rivet.
Buy a couple new #40 master links also. Use one to attach the 2 ends of the chain to each other after getting it wrapped around the sprockets.
Do you still have the old broken chain to measure or count the links you need?
My chain has 32 inner links.