Per State of Missouri: http://dor.mo.gov/faq/personal/indiv.php
If I am unable to file my return by the due date, can I get an extension?
If you receive an extension of time to file your federal income tax return, you will automatically be granted an extension of time to file your Missouri income tax return. Attach a copy of your federal extension (Federal Form 4868 or 2688) with your Missouri income tax return when you file.
If you expect to owe income tax, file Form MO-60 with your payment by the original due date of the return. An extension of time to file does not extend the time to pay. A 5 percent addition to tax penalty will apply if the tax is not paid by the original due date, provided your return is filed by the extension due date.
How to File for an Extension of State Taxes per TurboTax (https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Extensions/How-to-File-for-an-Extension-of-State-...