I ran into this problem and the solution required a rather technical fix that I am not sure many other Word users will want to do. I submitted a report on this to the Feedback Hub, but since it does not seem that the Feedback Hub and the community are connected (or at least, I do not see how they are) I am submitting the following solution to the community in the case that some other people may benefit from this research.
The current Chicago Manual of Style (sixteenth edition) states, "Two or more references in a single parenthetical citation are separated by semicolons." However, when I have attempted to use the Chicago style in Word for Author-Date citations, I note that the style uses a comma to separate multiple references in a single citation, rather than a semi-colon.
The offending line(s) in the file %APPDATA%\Microsoft\Bibliography\Style\CHICAGO.XSL are lines 2757 - 2759:
Note the use of templ_prop_ListSeparator instead of the more appropriate templ_prop_GroupSeparator. If you use the GroupSeparator instead of the ListSeparator you will get the semicolon instead of the comma for separating multiple references in a single citation. The result looks like this:
The way that I fixed this was to create a copy of the CHICAGO.XSL file and make the change their, and then use that new file instead of the old. If you do this, you will need to find the block testing for b:StyleNameLocalized (line 2124 in my version) and go through the names and change them to something other than CHICAGO or you will not be able to differentiate your copied file from the original file in the style menus.
I hope this helps someone and allows them to get the citations that they want without the trouble. I couldn't find any direct help on this online, so good luck!