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From what I know you will have to have the chapter to do antiquities. A lot like Jewelry Crafting and Summerset.
Like Jewerly crafting it will be introduced throughout most of the zones and that includes Cyrodiil. Its a lot like their housing update where they introduced houses in all the zones or most of them. Only difference here is the requirement to have purchased Greymoor.
This is also something that will be updated and expanded upon over time so its not just going to be abandoned. From what they have told us.
From what I understand, the quest giver for Antiquities is located in Solitude, so yes, you will need the chapter to start that skill line. They always lock the nice things behind the chapter, although we are not sure yet exactly how nice, grindy, or frustrating the new skills are going to be.