We had a new baby only a couple of months ago and do not have an SSN for them. Is this necessary for filing them as a dependent?

[Carl]

If the baby was born i9n the U.S. then you absolutely must have an SSN in order to claim them as a dependent. As I understand it, that baby couldn't even have left the hospital without the parent first having completed an application for an SSN. But then I don't know your situation. If the baby was born "a couple of months ago" then you should have received an SSN card for the baby by now.
If you did not yet apply for an SSN for the child, then you need to visit your local social security office on Jan 2 and bring all the paperwork with you from the hospital (or whatever institution the baby was born in) as you will need it. I do know that "some" of the paperwork required will be the original certificate of live birth with original signatures on it, along with other paperwork that I can't recall exactly, which you would have been provided prior to departing the medical facility where the birth took place.
Once you submit your application, you should receive the SSN card in 4-6 weeks, which gives you plenty of time to have it for tax filing, provided you don't procrastinate.

[VolvoGirl]

When you get the ssn and enter your baby in the program be sure to say he/her lived with you for the WHOLE YEAR for newborns.  

A child born (even on Dec 31) is considered to live with you the whole year.  So pick the Whole Year at the top of the drop down box.

[christianpatin]

Thank you so much for all the help, everyone!

[christianpatin]

And just for everyone's edification--Yes, hospitals do provide you with all of the documentation to have the SSN automatically ordered. Our first child was that way. But our second child was delivered by a midwife, whom cannot submit that kind of paperwork in California. So we had to do it ourselves.

[Opus 17 · Accepted solution]

If your child is eligible for an SSN, do not file your tax return until you receive the SSN, even if this means you must file after April 15.  If you file without the SSN, some tax benefits for your dependent will be denied, and they will not be paid even if you file an amended return after getting the SSN.

You must file your return with the SSN and on time; this means file before April 15, or get the automatic extension and then file before October 15.  If you file without the SSN, or you are late (you file after April 15 and don't have an extension) you won't get the tax benefits and credits.