How do I redesign web page without access to source code or API?

You can see the codes (HTML only) when you inspect the elements. F if chrome/firefox.

Other than that, you will still need the to access an administrator account either to the hosting site or FTP file management tool access to get the codes.

I am sorry . I may be wrong but if u created the website u should have the codes. Or if u have purchased it u should have someone helping u. 

Sorry but really I don't know.