I am writing a document in RStudio / Rmarkdown using Tufte style template. It creates pdf or html documents but is it possible to get the same output as a Word document?
No, or at least not easily as the inner workings as the style template are implemented explicitly for pdf and html.
You could look into adding this for Word yourself. I do not know how feasible that is.
I have forked the Tufte style and updated it to something a wee bit more modern looking (crisper font, no pale yellow background, ...) which I called 'tint' (for 'tint is not Tufte', gotta remain recursive). Its web page, the GitHub repo, and CRAN page all have some more information.