Is the military base overseas state (APO) available? I don't see it as an option to choose.

Change the address TYPE to APO/DPO/FPO.   There is a drop-down menu where it says US Address.   

We moved from overseas to the state in 2024. My child received the childcare service on base overseas as the same in 2023. When I tried to claim the childcare tax credit part. The information was saved from 2023, but not the state part. There's no state option for "AP," but when I chose "CA," it usually works for AP address. It keeps giving me an error saying, "The state doesn't match with the zip code." I cannot move forward unless I solve this issue. :'(

If you are in the review, exit from there and go back to your entry in the Deductions & Credits section.   You need to use a CA zip code if you are going to use CA for the state code.  

Check the 2-letter state code for that zip code.   Make sure the city and state match up to the zip code.  This error happens when the state code doesn't match what is on file with the IRS.  TurboTax follows the IRS for acceptable zip codes.  If the state code does not match the correct state, you will get an e-filing error.   Correcting the state code is usually the fix.  

• Check that the 2 letter state abbreviation is correct here USPS Two–Letter State and Possession Abbreviations   (most common error)
• Make sure you've entered the ZIP code in either 5-digit or 9-digit format, for example, 12345 or 12345-6789.
• Look up the correct ZIP code for the street address in question.