I'm wondering if I need to send the original W2s (which are at home) or whatever copy I'm able to print off TurboTax--which apparently is a copy of sorts.
It is certainly okay to mail exact photocopies of the (original) paper W-2s, or Form 1099s, that a taxpayer may have in their records. Although original documents are preferred by the tax authorities, photocopies will do just fine, as long as they clearly show your income tax withholding to the relevant taxing jurisdiction. What this means is that the W-2 you send to the IRS needs to clearly show the amount of federal income tax withheld in Box 2. The Illinois Department of Revenue will want to see the state income tax withheld in Box 17 of Form W-2.
That said, we would not recommend that you use a TuboTax screen print-out of your W-2, or any other tax document for that matter, unless it is an exact photocopy or scan of the original. The reason for this is that, while a revenue authority may accept a non-exact copy, there is no absolute guarantee that they will. Hence, if they do not, it could take weeks of waiting to hear from the government, only to have them send you an inquiry letter requesting additional information (original documentation), rather than sending you a tax refund. Thus, to avoid what could be, potentially, a significant delay in the processing of your tax return, it's a wiser course of action to take some extra time in the beginning, and provide them what they want in the first instance . . . to avoid any unnecessary future delays.
One additional point that you (and anyone else reading this post) may find interesting. Have you ever wondered why the W-2 you get in the mail from your employer has three or four separate copies of the identical information? Well, that's because these tax forms are a "holdover" from the days before e-filing, where taxpayers always had to file their tax returns by mail, as that was the only option available to them. The W-2, and other similar tax forms, usually have perforations punched into the paper, and were meant to be torn apart, with one copy kept for the taxpayer's own records, the second copy to be mailed to the IRS, the third copy to be mailed to the state, etc. You may or may not have know that fact already, but it's an interesting (historical) observation on the evolution of the income tax filing system in the United States.
Thank you for asking this important question, and good luck to you.