I'm active duty military and my home of record is in Ohio (where I pay state taxes) My wife pays state taxes for Missouri. I cannot finished my taxes due to that issue.

If you file a joint federal return, Ohio requires you to file a joint state return. List yourself as resident and her as non-resident.

Ohio does a convoluted tax calculation for non-residents/part year residents. It calculates tax on total income, then it calculates a non resident/part year resident credit, which it subtracts from the tax it calculated on the total income. The credit is calculated as your non-Ohio income divided by Total adjusted Income multiplied by the total tax. TurboTax (TT)   does this by allocating your income as either Ohio or non-Ohio. W-2 income will be allocated by the state name abbreviation shown in box 15 of your W-2. TT will ask you, item by item, in the state section, how much of your other income is Ohio or non-Ohio income. Make sure that her non-Ohio wages show MO (Other state postal abbreviation)  in box 15 of your W-2 screen, with the MO amount in box 16. 

If your only income is military salary, and it was earned outside Ohio, you will owe no tax (the credit will be 100%).