There was no Mordor (as far as we can tell) in the First Age. In the Second Age Sauron set up shop there pretty early on. Well, okay, it took a thousand years, but still...
If anybody lived there it would probably have been tribes of primitive men who migrated in after Mordor was created. (And if they had any sense would have left when Sauron turned up. I wonder if they could have been the original slaves who worked Sauron's fields in Nurn...)
The dwarves mostly lived pretty far to the north. Durin's Folk lived in the Misty Mountains, the Grey Mountain, the Iron Hills, and maybe Erebor. This area would have provided them with plenty of exploitable resources for a long, long time (as it indeed did).
The Firebeards (Nogrod) and Broadbeams (Belegost) were probably spread up and down the Blue Mountains during the First Age. Not only did Mordor not exist but it was well beyond their range of activity. After the First Age their populations were probably shattered (the Firebeards certainly were) and the survivors probably remained in the traces of the Blue Mountains or went to Khazad-dum.
Nothing is know about the other dwarves except that they lived far to the east. I think it unlikely they had anything to do with the place.
I suppose houseless Elves might have passed through, but...
This may be another reason why Sauron liked the place. There weren't too many people there to bother him.
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