With a minimal understanding of IP and copyright law, combined with a glance at the ESOUI terms of service, it should be obvious that nobody is entitled to access the API. Simply put: ESOUI is private property and not yours or mine to take (in this context, "access" = "take").
If you want to create a new client interface, you should contact the owners of ESOUI directly (not by posting in these forums) and make a proposal. They own something that you want. Make an offer they can't refuse. Heck, if your pockets are deep enough, perhaps you can buy the entire website from them. Good luck!
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