This post suggests that the engine you have is an 143.676222:
According to my Tecumseh Technician's Handbook, 143.676222 is a Tecumseh HM80-155129D. Judging by the aluminum recoil starter and the choke knob by the flywheel cover on the left, this is probably a late 70s vintage snowblower. If you remove the electric starter switch (#3 below), you will likely find the model number stamped underneath it:
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If it is a Tecumseh 143.67622, the carburetor for that engine is 631957B. That carburetor is still available new, but as a "service carburetor." What I mean by that is you'll need to reuse some of the parts from the original carburetor like the choke butterfly valve and shaft. Here's a pic of a 631957B:
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It looks to be similar dimensionally to the ubiquitous $12-15 aftermarket carburetors for 8-10 hp Tecumseh engines, p/n 632334A:
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Take a look at your old carb and see if you'd be able to remove the choke shaft and butterfly to reuse it. If so, I'd start with an aftermarket 632334A and see if you can swap out the choke parts.
For what it's worth, I occasionally convert those 70s HM80s to the newer 90s throttle and choke lever design, if I find that the engine has good compression. Here are a few that I've converted:
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The benefit of this conversion is that you no longer have to rely on a remote mounted throttle lever, and the cheap adjustable carburetors drop right in (you will have to drill a hole on the carburetor cover for the choke shaft to poke through):
This 924039 now has the spare wheels with X-Tracs installed. The rims themselves are a bit crusty, but they hold air and the X-tracs have had minimal use under my ownership. I may install a taller chute at some point, and I still need to free up the differential lockout hub.
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