Yes there were, if we simplify the question to "people digging at old sites to uncover history" then the earliest evidence we see of this is at Catalhoyuk (ca. 9-8kyo), as Ian Hodder notes:

...in Building 1 retrieval pit F.17 was dug [in ancient times] to remove or retrieve a relief sculpture (only traces of which remained on the wall)...Given the large amount of erosion off the top of the mound that occurred in the millennia after the Neolithic occupation, we cannot know how deep these Neolithic 'archaeologists' had to dig, but it was at least 0.7m and probably substantially more. We do know that Building 1 had been filled and that any digging down implies a precise historical memory even if embedded within wider knowledge about where important sculptures were generally placed. Not all houses have major relief sculptures on the west walls of main rooms [where the retrieved sculpture was].

The earliest evidence of "organized" ancient archeology was done under the supervision of neo-Babylonian kings. They had "field directors," they did yearly excavations specifically searching for "lost buildings" from records, they uncovered and restored foundation deposits (in ancient languages), and sometimes kept the ancient objects they found as heirlooms. Over in the Americas, in North America during the Mississippian period there's evidence of people (likely priests) digging into old mounds to a specific level, leaving a votive deposit, and covering it up. While we don't know how "organized" this was, it is de-facto archeology as well. The last example I've spotted is from the Book of the Hopi, which gives the story of a Hopi man named Lesso who in the 1890's conducted indigenous archeology at a Ancestral Puebloan site.

There's a fine line between "archeology" and simply looting, and of course the latter would've been common in the ancient world. Adrienne Mayor gives examples in her book about Greco-Roman fossil traditions of that likely normal situation - a boneyard is uncovered somehow, usually from natural forces, and then it's looted by local peasants. There's also a few examples I've noted around the world where individuals have dreams that there is "something you want" in a burial mound, and then acting on that dream dig at the mound and find ancient objects.

I've talked about how ancient societies conceptualized their history (along with the various examples of ancient archeology) in this lengthy post, and I give some more details about archeologists and fossils in this post. And an important addendum, the notion that one should dig and uncover lost objects/buildings presupposes the idea that ancient buildings are immortal and history is linear (in some way). This is a very neolithic train of thought, and is a radical departure from indigenous values that buildings should fall back into the earth because they are mortal and history is cyclical. I've talked about the emergence of this implicit logic which undergirds both ancient and modern archeology in this post.

There's always more to be said on the topic, but I wrote about this in Were there archaeologists and museums in the ancient world? and Were there any archaeologists in ancient cultures?