Can I still use TurboTax if my wife and children do not have SSN or ITIN?

Yes. If you are considered a resident alien for US income tax purposes, you can prepare your income tax return using TurboTax. However, you will not be able to efile your income tax return if you are applying for an ITIN and including your tax return with this ITIN filing.

Please note : If you are on a F-1 visa, you will be exempt from the Substantial Presence Test for the first 2 (if non-student) or 5 (if student) calendar years that you are in the US. This means that you are considered a non-resident aliens for US tax purposes. Click this link for more information from the IRS on US Tax Guide for Aliens

Next,  you will actually file for your wife and children's ITIN and attach a copy of your federal return to your ITIN filing.

So, if you want to include your spouse and children on your tax return and they do not have any SSN or ITIN, you will need to apply for an ITIN for each family member. You would use Form W-7 and would need to attached your tax return to this form (see below). Please note, you will not qualify for the Earned Income Credit (EIC) without a valid Social Security number (not an ITIN) for yourself, your spouse (if filing jointly), and any qualifying children. 

Please click here for more information about Claiming Non-Citizen Spouse and Children On Your Taxes.

Also note that if your spouse is a nonresident alien (NRA), you will need to attach a statement, signed by both spouses, to your joint return for the first tax year for which the choice applies (Nonresident Spouse Treated as a Resident). It should contain the following information:

•  A declaration that one spouse was a non-resident alien and the other spouse a U.S. citizen or resident alien on the last day of your tax year, and that you choose to be treated as U.S. residents for the entire tax year.
• The name, address, and identification number of each spouse.

Here are some additional steps for filing married filing jointly if spouse and family needs an ITIN. 

• leave spouse and dependent's SSN blank and print a paper copy of your tax return.
• skip error check section if only error is missing SSN for spouse and dependents.
• under the file tab, you will need to select "file by mail" and print a copy of your federal income tax return.
• apply for spouse and family’s Individual Taxpayer Identification Number (ITIN) using Form W-7 (Click ITIN). (You will need to apply for a ITIN if filing married filing separately and taking an exemption for spouse and that spouse does not have a SSN. (see below))
• attach a copy of your Form W-7 (along with the necessary supporting documents) to your tax return (For the supporting documents, one option is for your spouse and family to get a certified copy of his/her passport which will need to be included with your tax return filing.) Click here for IRS website  ITIN - Frequently Asked Questions .
• mail your tax return along with your Form W-7 and supporting documents to the IRS. According to the IRS, because you are filing your tax return as an attachment to your (ITIN) application, you should not mail your return to the address listed in the Form 1040, 1040A or 1040EZ instructions. Instead, send your return, Form W-7 and proof of identity documentation to:
  
       Internal Revenue Service
       Austin Service Center
       ITIN Operation
       P.O. Box 149342
       Austin, TX 78714-9342 
  
  You may also apply for an ITIN using the services of an IRS-authorized Acceptance Agent or visit some key IRS Taxpayer Assistance Center in lieu of mailing your information to the IRS in Austin.

• State filing - You will want to file an extension for your state until you receive your ITIN. This is because the application for your ITIN is a federal filing. Most state will not process your return without either a SSN or an ITIN. (However some states will accept state filings with W-7 attachments. Please contact your state for more information using the State Department of Revenue Contacts and ask if your state will accept a state tax return with a Form W-7 attached).  
  When the IRS assigns you an ITIN, update your returns (both federal and state) in TurboTax with this new information and then you should be able to e-file your state return.
  Because many states give an automatic extension when you file the federal extension Form 4868, just file a federal extension even if you don't think you will need it for your federal tax return.  But check with your state's Department of Revenue to make sure that your state doesn't have a separate state extension filing.

Great! Apart from the fact that Turbo Tax keeps asking to type SSN for my dependents, it works great.

Do My kids qualify for child tax credit? They are not US citizens or permanent residents. Turbo tax grants them child tax credit and I could not take that away.

Yes they could if they get an ITIN and they are treated like US resident aliens for the entire 2016 tax year (along with meeting all the other requirements). According to the IRS, to meet the citizenship test, the child must be a U.S. citizen, U.S. national, or U.S. resident alien. Please click this link for more information : <a rel="nofollow" target="_blank" href="https://www.irs.gov/uac/ten-facts-about-the-child-tax-credit">https://www.irs.gov/uac/ten-facts-about-the-child-tax-credit</a>

Hello,
I've followed the instruction, downloaded and printed Form 1040, and filled W-7 forms for my wife and children and filed all the documents in IRS Taxpayer Assistance Center.
The ITIN application for my wife seems to be accepted, but the applications of my children were rejected. I've received letters from IRS with the text:
"We rejected your application for ITIN because you didn't include a completed and signed U.S. federal income tax return with your application."

I've listed my children in the "Dependents" table. For each of the children, I've s I left the "Social Security Number" field empty and I didn't mark neither "Child tax credit" checkbox nor "Credit for other dependents" checkbox. Is it correct?
I'm not really sure what's wrong with my Form 1040 and what are my next steps. 
Thank you for the assistance