Do I have to mail in my earned income tax papers to the state of Vt if I e-filed?

No.  You have to wait for federal to be Accepted first, which won't be until the IRS opens on Jan 29.  The state won't be efiled or accepted until federal is first Accepted.  

And e-filing for VT isn't scheduled to be ready until after 25 January..

......so don't try to file until the IRS start.   

If you have already e-filed the Federal part, you will have to come back later... After the Fed is accepted...and AFTER 25 January in any case to e-file your VT forms.

If you e-file VT, no, you don't have to send in your income forms separately...BUT,, you need to keep them in a labelled file folder , because occasionally, the state askes taxpayers for paper copies later if they think something is "off"....or even an occasional random check.