Building a local database from Web of Science and Scopus search results
Author disambiguation is a big problem in most plattforms / citation databases, and an issue in much bibliometric work. I don't think you will find a quick fix. Orcid (which WoS sometimes provides) is aimed to partially solve this, but depends on researchers actually having and using Orcid. Scopus currently has a better functionality in finding publications from standardized author records (i.e. "all" publications for an author, after their algorithm for merging authors have been applied), and this is also exported in e.g. their CSV format in the field "Author(s) ID".

See for instance these exported records from Scopus, with overlapping authors (and therefore author-IDs):

'Author(s) ID'  EID
24802234300;6602122539; 2-s2.0-84861411180
24802234300;55923304700;6603273826;6602122539;  2-s2.0-77955097159
16229637800;6602122539;24802234300;7003423871;  2-s2.0-51749084398


So if you want to build a local database from an online bibliometric source, which include some form of unique author ID, I think Scopus would be a better starting point. Web of Science has their own identifier included in exports (ResearcherID in the RI field) and Orcid IDs (OI field), but only for researchers that has those IDs, which is usually the minority (both across papers and within individual papers). One downside with Scopus is that there export does not include Orcid from what I know (at least not in the open web interface or the basic API), so it may be harder to integrate with other data sources. 

Note however that all identifiers are not always included in all export formats (goes for both Scopus and WoS) so you should focus on e.g. csv-exports (which include the identifiers I mention above).