I don't know what the best place is to report an issue like this so I turned to this forum.
PROBLEM: My character on Frostmourne, Enhonc, has 1100 Achievement Points but it doesn't count towards getting Points on the site. When I press "Collect points" it says I have no eligible characters, and my activity rank is permanently on the lowest one.
Correct. There is no point shop on Frostmourne, so if you only have characters on Frostmourne you will not be able to collect points. However if you have a character on Icecrown with at least 1k achievement points, you will be able to collect points. You don't need to log in to the Icecrown character, it just has to be on your account.
I assume this is the same if you have an account with characters only on Lordaeron, you wouldn't be able to collect points because there is no point shop.
However I think they should enable the collect points feature on account of Frostmourne characters will transfer to Icecrown in a year - so why not let people collect points in the meantime even if they can't use them yet? Then again they're mostly used for catch-up gear for a fresh 80, so who knows.