Starter on snowblower won't engage

Need to know what year you have. If the starter solenoid is not activating the starter drive or jack shaft then the starter would spin freely and whine. If you have a bent shaft or otherwise that will stop gear engagement when you try to start it would just spin and whine. SO, if you get a whine the starter is spinning but not engaging.

If the starter relay makes a click everything up to there is working.
If the solenoid makes a click everything up to there is working ie the low current circuit is working. If the starter spins the high current circuit is also working and that leaves the above that the starter drive is not working to engage the starter with the clutch ring gear.

A common problem with snowblower starters. Rust gets all around the kick out pinion gear. When you apply power the throw out gear spring or mechanism is rusted an will not operate. Take it off engine an check for broken teeth as well. Most often is rust on starter clean if you can an lube but usually there is no fix

take starter out and clean small shaft.. Will it start with the pull starter?

Return starter bc the starter solenoid is not engaging bendix, which engages the starter teeth into the flywheel..

The starter on your bike uses a solenoid that also manually engages the starter gear. Try lubricating the solenoid mechanism and verify that the mechanism does not bind or stick. Once you have the mechanism freed up, check the starter again. If it does not engage, replace the solenoid (or, while you're at it, just replace the entire starter assembly).

Yep, not enough amps to engage the starter clutch, and when ti does engage, not enough amps to turn the engine. You need a 12 amp battery.

check your flywheel & starter, one or both will be missing teeth, or the starter isn't "throwing out" yo engage flywheel

take it off and see if the gear is broken (you can buy the gear at napa)or did the shaft the gear rides on got a little rusty over the winter