Do I need to file an amended state return to get my refund for unemployment in Missouri? Will doing so have any effect on my federal unemployment return for the $10,200?

State Taxes on Unemployment Benefits: Unemployment compensation is generally taxed in Missouri to the same extent it's taxed under federal law. As a result, the state has adopted the federal exemption for up to $10,200 of unemployment compensation received in 2020. If you previously filed a 2020 Missouri tax return and reported unemployment income that is now excluded from your federal adjusted gross income, you must file an amended Missouri return to claim the exemption for state tax purposes.

"Will doing so have any effect on my federal unemployment return for the $10,200?" not sure what you are asking. you did not file a federal unemployment compensation return.  you filed a 1040 on which you should have reported all the UC you received. if you itemized your deductions for federal purposes. it is possible that part or all of the state tax refund will be taxable for federal purposes in 2021.  the federal refund is automatic if you qualify but you may not receive it for months. . the IRS does not want amended returns filed to claim the UC exemption.