Im in the U.S. since August 2015 on O-1 visa, my wife is on O-3. She isn't eligible to get SSN, and didn't obtain an ITIN. How can we proceed filling out the declaration?

As a O-visa holder, during 2015, you were considered as a nonresident filing form 1040-NR due to not meeting Substantial Presence Test. However, if you anticipate you will be residing in US for at least 183 days in 2016, you can make an election to treat yourself as a resident filing form 1040 from August 2015. FirstYearChoice- under Residency Starting Date under the First-Year Choice

You cannot file Form 1040 or the statement until you meet the substantial presence test in 2016. If you have not met the test by April 18, 2016 , you can request an extension of time for filing your Form 1040 until a reasonable period after you have met that test. To request an extension to file until October 15 of the following year, use Form 4868, Application for Automatic Extension of Time to File U.S. Individual Income Tax Return.

If you decide to make the election, you can also make another election to treat your nonresident spouse as a resident. Nonresident Spouse treated as resident – page 9  You both will be filing form 1040 Married Filing Jointly.

If you decide to file form 1040-NR, you cannot use TurboTax to file.  You are suggested to use Sprintax.

If you decide to make elections to file as a resident, you can still use TurboTax to prepare your form 1040, then you will print and mail it out to IRS.

Does your spouse have a social security number or ITIN ?

If not, you will need to apply for one https://www.irs.gov/Individuals/General-ITIN-Information, print form 1040, attach it to W-7 with the required documentation and mail them together to the designated IRS location at Austin TX. 

Thank you for your response! It's very helpful. Although, I'm a little bit confused with this substantial presence test in 2016. Do we have to reside in the U.S. for 183 days in 2016 starting from January 1 2016, since we didn't leave the U.S. at all in 2016, and only then file the return (on July 1st)?

If you want to make the election to claim yourself as a resident, in 2016, you will have to use the formula ( total days in 2016 + 1/3 of days in 2015 = at least 183 days)  to determine if you need the extension or not.

Thank you for your response! One more question. In 2015 we were on tourist visa since the beginning of the year till august and then changed our status to O-1(work visa), O-3 visas. Should we count whole year or only that part, that I was on work visa?

For tax purposes, doesn't matter if it's the tourist or work visa, as long as it is not the exempt visa like A or G, your days being in US count for meeting the Substantial Presence Test.