Can I file taxes for another person?

If you're preparing multiple returns for family members, it's more economical to switch to the TurboTax CD/Download software, which lets you e-file up to 5 federal returns at no additional cost. More info

Instructions on how to start another return online:

1. Sign out of TurboTax Online.
2. Select the TurboTax Online product you want to use for the second return and proceed.
3. On the Create your account screen, set up a new login for the second return.
   
   • You can reuse your email address if you want to receive e-file notifications for the second return.
   • Up to 5 accounts can be tied to the same email address.
4. After you've created the new account, you can start working on the second return.

Will I be charged a fee for completing another return even though I paid a fee for federal and state when I prepared the 1st return?

@PT735   It depends what product you are using.    If you are using the Online web  version it is only good for 1 return per account.  You have to pay for each return.  You can use the same email address for 5 accounts.

How to start a new online return
https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-online/00/25596

If you bought and installed the CD/Download Desktop program it can do unlimited returns and efile 5 for free.  Each state is $25 to efile or you can print and mail state for free.  And the Desktop program has several other features over the Online version.