Whether you can claim dependents doesn't have anything to do with a 1099-MISC or a W-2. A self-employed person (1099-MISC) or an employee (W-2) can claim dependents. A lot of self-employed people have dependents.
Maybe you're thinking about which of the income tax forms you need to use, i.e., 1040EZ, 1040A, or 1040.
With dependents, you cannot use Form 1040EZ. You can file dependents on either a Form 1040A or a Form 1040. But you cannot use Form 1040A with a 1099-MISC, so that means you have to file a Form 1040.
TurboTax will know which of the Form 1040's to fill out for you based on your entries.
In the Online products, Deluxe and above can prepare a Form 1040. However, Deluxe may or may not be able to handle your 1099-MISC. Online Deluxe and Premier can only handle a 1099-MISC in some very simple circumstances, such as if one has expenses less than $100.
A 1099-MISC for a self-employed business will ultimately show up on a Schedule C or a Schedule C-EZ. While Online Deluxe and Premier can prepare a Schedule C-EZ when people qualify for that simpler form, they cannot prepare a full Schedule C for a self-employed business/independent contractor. In an Online product, only Home and Business can handle a full Schedule C for a self-employed business.
Here's the difference between Schedule C-EZ and Schedule C:
https://ttlc.intuit.com/questions/1901317-how-does-schedule-c-ez-differ-from-schedule-c
