The Dropbox API does offer the ability to programmatically search in the connected account, with a number of different options. You can find more information in the documentation here:
https://www.dropbox.com/developers/documentation/http/documentation#files-search
You can specify the query, such as a file or folder name or file extension, that you want to search for you in the "query" parameter.
You can also restrict the search to a particular path using the "path" parameter under the "options" field, specifying either path or ID.
It's not currently possible to request non-recursive search, but I'll pass this along as a feature request. I can't promise if or when that might be implemented though.