The issue is likely to be about incomplete account information.
A new Apple ID account has a few requirements:
- The email address used has to be verified (respond to message);
- The account needs its user details. Name, address, country, DoB, phone number, payment method (for optional non-free content or apps).
I don't have a computer either.
😯 That might be an issue. See how much you can set/complete at the Apple ID account site (e.g. on a smart phone or tablet — you do have one of those, I presume?).
With all the account requirements met, sign in again to the Apple TV box with this account (Settings﹥Users and Accounts﹥), and the App Store should be accessible then.