Yes, you have to buy the federal software. The state software does not run by itself. It's an add-on to the federal program.
Furthermore, the state tax return is based on the federal return. If you did not file your federal tax return with TurboTax you will have to recreate it before you can start the state tax return.
In addition, you will need both state programs because there is an interaction between the two states. You will get a credit on your Louisiana return for part or all of the tax you paid to Arkansas. So you also need the Arkansas return in order for the Louisiana software to calculate the credit.
If you have a .tax2021 file for your federal and Arkansas returns, you will not be able to open the file until you have first installed the Arkansas software.