Can you obtain snowballs without a shovel?

Solution 1:

Yes!

0-15 snowballs are dropped by snowmen.

Why does that help you? Snow golems require snow blocks, but snow blocks do indeed occur naturally, even two on top of each other, in igloo walls, and two next to each other (can create snow golems as well), in ice spikes biomes and both in frozen ocean biomes.

And the pumpkin? How normal pumpkins generate naturally is pretty known, but for a snow golem you need carved pumpkins. But they do indeed also generate naturally! Quote from the Minecraft wiki:

Carved pumpkins generate in rail rooms in woodland mansions.

Carved pumpkins also generate in pillager outposts as a part of targets/scarecrows.‌ ^{[upcoming: JE 1.14 & BE 1.10]}

Endermen can pick up and put down pumpkins, not even considering that dirt underneath is needed. So that means that in theory an enderman could randomly pick up a carved pumpkin from a pillager outpost or woodland mansion (already pretty rare and the generation would have to be pretty special to even allow an enderman to do that) and then he could randomly wander around, either towards an igloo or a special formation of snow blocks in an ice spikes biome or, even less likely, an iceberg in a frozen ocean biome (in the middle of lots of water, but theoretically possible) and then he could randomly decide to place down the carved pumkin in exact the right location, creating the most unlikely snow golem in the history of Minecraft… only for you to come along and slaughter it. But remember to not do it by punching, otherwise your challenge is over.

So yes, it is possible to get snowballs without left clicking, but it's very hard and insanely luck related. What you could do alternatively is using my control scheme: Left click to place, right click to mine. ;)

Solution 2:

The answer is yes, you can obtain snowballs without a shovel. In fact, it is even possible without left clicking (though it is fairly difficult and will likely take multiple tries).

Here are the steps to do it:

1. First thing first, you need to find a village that has a farmer, a library, and an iron golem.
2. Because farmers sometimes sell pumpkins, you can trade emeralds for pumpkins with the farmer (emeralds are obtainable without left clicking).
3. In 1.13 onward, snow golems require carved pumpkins rather than normal pumpkins to create. Do this by killing the iron golem and crafting the iron ingots into shears using the library's crafting table. Then you need to blow up the pumpkin, and hopefully it will drop.
4. Next, you need to find a snow biome with two blocks of snow stacked on top of each other. Then, of course, place the carved pumpkin on top to spawn the snowman.
5. Kill the snowman. One of the best ways to do this might be throwing eggs, because they are relatively easy to find in most worlds and don't require left clicking.