it was designed by the US branch of the Mackay clan, who have attempted to get our Chief, the Lord Reay, to recognize it but have yet to get any response from him, so it was registered with the STA as an unofficial and a corporate tartan of the Clan Mackay Society, USA
I read this to mean that the tartan was designed (and it wouldn't be unprecedented -- most "dress" versions of clan tartans began as fashion designs that were later on adopted by the clan). Then the design was submitted to the chief for his approval.

I'll stop here and say that I think up to this point all is in good form. One could argue that they should have sought the chief's approval before creating the design, but one could equally argue that getting the chief's approval would be difficult without a design for him to either approve or disapprove. I can see both points of view, and I don't think anyone should call foul for someone holding the other.

In any case, the tartan was designed, and then the chief's approval sought for the new design. And I read the above to indicate that the chief declined to either accept or reject the new tartan. And I also read the above to imply that a signifigant amount of time was allowed for the chief's reply.

No reply forthcoming, either positive or negative, what do you do with your new design? It would be improper to record it as a clan tartan without the chief's approval, so you record it as a corporate tartan for the corporate body you represent, in this case the US Clan Society. Again, not unprecidented. Many US Clan Societies even have their own heraldic Arms and crests that are recognized by the Lord Lyon. Why can't they have their own tartans, as well?