I'm with Garandman on this one 100%. Sandblasting would be a disaster. The sand (you shouldn't use sand because of the possibility of silicosis from fractured sand) will probably be what is called "black beauty" is really silicon carbide. Pretty inexpensive. Sandblasting is also relatively slow if you have a lot of surface to clean.
However, unless you have a cabinet that fully encloses your workpiece, when you blast, the grit gets EVERYWHERE. In your hair, on your clothes, and all over everything within several feet of where you're working. Then anyone that walks into your area then tracks it into the house.
It will also get into whatever bearings or bushings you may have in your housing.
Depending on machine age, your luck, and/or condition, taking the thing apart can be easy or very difficult and might require special tools. I don't know much about your machine, but if you can get to the back of the housing, there is probably a pulley there. If you can get that off easily (and any other things that might be on that shaft), then that's probably the hardest part. There's also probably a bearing or bushing retainer that holds the shaft in place. The impeller and auger assembly is now basically supported in three places- the rear bearing mount and the two sides of the auger. If you can disassemble these three points, then you should be able to jockey the arrangement out.
If you get the impeller/ auger assembly out (and any bearings or bushings), then you COULD sandblast. It will still take quite a while. I used paint remover on the housing on my '88 Ariens. It also took a while, but it was much cleaner and I could do it at my convenience. After getting most of the paint off, I used a spray Rustoleum rust converter product (from Walmart or similar) as a primer. I'll leave the finish color up to you.
Other pieces of advice: 1. Take lots of digital pictures along the way- you won't remember how some things go back together. Especially important if you do step three below. You won't remember which auger rake went where.
2. Keep all parts together in a box or bag, else you will lose something.
3. This would be an excellent time to make sure the auger rakes spin free- remove the shear pins and grease the shafts before reassembling.
4. I didn't see where you live, but this might be a little late in the season for snowblower disassembly and painting.
Hope I didn't forget anything. Good luck
Ralph