G'day Basile:
I've been (and still I am) with this kind of issues when one inherits other's protocols, structures of folders etc...
With the information you provide (and keeping the solution inside R), I reckon I would load the document into R, probably with readLines, and then search for the lines that have some regular expressions like read.csv, readRDS, or anything like that, and see what is being read / imported along the script.
However, you mention hundreds of files, so there may be files loaded/called inside files that then are called in/loaded in subsequent scripts... a real mess.
Such a great mess.
In your case, I would do it by hand. Separating the Rmd file form the other files and in a clean environment sequentially run it and find out what's missing at each step. I don't say don't use R or anything "programming related", I say for me it would be faster to do that way (few hundreds of files = a long couple of hours?) vs many hours finding a solution inside R and making test to prove that it is working safely.
I'll keep an eye to see better solutions, as it interests me 
cheers
Fer