Is the US Military Day Care considered a tax exempt organization?

Child Development Centers (CDC) operated by the Department of Defense are government activities and are exempt from tax, i.e. "tax-exempt".

So, does that mean where you are supposed to input child care, you can not put military child care centers ? 

No, you should still claim child care expenses, for military child care centers, but you won't be able to e-file your return. 

You'll need to actually print our and mail your return, and write "EXEMPT" where the childcare provider's TIN would normally be, since the IRS won't accept an e-filed return with "EXEMPT."

You might also try the IRS's page at Tax Exempt Organization Search to search for the provider there, just in case it's listed.