How can I collect all the Snow Globe Trading Cards with many restrictions on my account?

The "restrictions" you're saying you've got on your account do not match up with what Steam call a "limited account" - it is probably worth contacting Steam support to discuss the issues you're having with the Steam store.

There are four methods of earning Snow Globe cards:

• Vote in the "Community's Choice" sale every 8 hours, for every three times you vote you get one card (that's one free card per day)

  

• Buying games/software from Steam: for every $10 you spend on Steam you get a Snow Globe card

• Trading with other players on the Steam market

• Crafting other badges will result in a Snow Globe card for each badge crafted

One thing to note is that getting Snow Globes by voting for the "Community's Choice" sale requires that you be level 5 already.

You will have to spend much more than $100 to get all 10 of the snow globe cards, as you will get cards randomly from buys. This means that you will end up with multiples of the same card, and not get any of the other cards you actually need. If all you can do is buy games, it will likely cost you a very large chunk of change if you want to max out the steam snow glow badge. It will likely take over $300 just to get level 1 badge. And the only way to get the foil badge would be to buy them from the marketplace, as they are so rare.

The only economical way to achieve your goal is to get Steam support to disable steam guard, or set up a new account, buy the cards you need, and trade them back to your main account.