Yes, you can contribute to your 2016 Roth IRA from now until 4/18/2017. Be certain that your custodian makes the contribution for 2016 and not 2017.
Note: Because Roth IRA contributions are made with after-tax monies, they do not have to be entered in TurboTax at all. It is just a handy place to keep track of Roth basis for future distributions. There isn't a line entry for Roth contributions on your tax return like there is for a Traditional IRA, because there is no deduction for it.