Sorry no. You have to send your copies to the state. And Turbo Tax doesn't have your actual W2s, they only have the info you entered from them. You need to keep your own W2s. Turbo Tax does not deal with the IRS or state for you. Turbo Tax is a self prepared return.
No, TurboTax does not have nor can access the W-2's for you.
Only your employer's can provide the W-2's.
If you cannot obtain the W-2 from your employer, then for a fee you can get a complete copy of a W-2 from the Social Security Administration - https://faq.ssa.gov/en-us/Topic/article/KA-02501
TurboTax (TT) does not have your W-2 (in either the on-line or desk top versions). However, it does have a "facsimile", if you typed in (or imported) your W-2 when you prepared your return. If all you need is the information from your W-2, you can get that from the facsimile (W-2 worksheet), found in the list of forms. Every bit of information on your actual W-2 will be on the TT worksheet. Whether whomever you need to show your W-2 to will accept the worksheet instead; you'll have to ask them. Your employer is the best source of an actual copy of your W-2. For more options, see:
https://ttlc.intuit.com/community/taxes/discussion/how-do-i-get-my-old-w2s/01/782516#M311511
https://ttlc.intuit.com/community/after-you-file/discussion/i-need-to-get-a-copy-of-my-2017-w-2-where-can-i-find-it/01/789150#M163978 link Includes:
IRS instructions http://www.irs.gov/taxtopics/tc159.html
Social Security Administration Instructions questionID=3836&hitOffset=258+164+163+75+26+25+15+14&docID=2040
it seems unlikely that Oregon will accept a TurboTax printout of the W-2 information in the filed TurboTax state tax return.
But, what do I know, I could be wrong.
cc: @hazelmist