I made a max contribution of $6500 to my wife's IRA in Feb 2016 to offset my 2015 tax liability. However I just found out 1/4/17 when I tried to make a contribution for the 2016 tax, that Vanguard had applied my 2015 contribution to the 2016 tax year. This not only makes us ineligible to contribute for 2016, it makes my 2015 tax return inaccurate. How can this be rectified?
I agree with @jeffelm1's answer. If it was Vanguard's error and you checked the 2015 box on the contribution form (if they gave you one to fill out), then only Vanguard can reissue a corrected 5498 form to report to the IRS a change in tax years for the contribution.
By default, contributions are for the current tax year. It's much too late for Vanguard to change the year designation of the contribution unless perhaps you have a copy of documentation that you provided to Vanguard with your contribution indicating that the contribution was to be for the preceding tax year, in which case they can treat it as a bookkeeping error and issue a corrected 2015 Form 5498. Had you noticed this error before the regular filing deadline for your 2015 tax return, they probably could have made the correction regardless.
Assuming that the year designation cannot be changed, you'll have to amend your 2015 tax return to remove the contribution for 2015 and report it as a contribution for 2016 on your 2016 tax return instead.