Is there any way to find all papers that acknowledge a specific author? (i.e., the author is mentioned in the acknowledgement section)
Resorting to advanced search tool that most famous publishers offer would simplify your task (at least to an extent).
E.g., search for papers with name in full-text and not in authors and not in references.
Currently existing journal libraries do not support such a feature yet. The solution to your question is a topic of research by itself. The following articles,  and , might provide some insights.
As recently as February 2015, there was such a search engine, called AckSeer (patterned after CiteSeer): http://ackseer.ist.psu.edu. Unfortunately, it seems to be inactive now. You can see what it used to look like on the Internet Archive. There is a scholarly article that describes it in detail.
Now, probably the only way to do search for authors in acknowledgements would be to search in a scholarly database that lets you:
- Search the full text for the desired author's first and last name in quotes;
- and simultaneously exclude the author's name from the author search.
Searching for the author's first and last name in quotes (e.g. "Firstname Lastname") is probably the best way to avoid returning articles that mention the author in the reference list, since most reference lists don't list authors in that format. This suggested technique is not foolproof, but it's the best way I can think of to get the closest to what you're asking for.