Someone who knows how to fix this problem:

Code should be placed in three backticks as shown below

annot <- read.csv("samplesheet.txt",header=T,sep="\t")
annot$Slide <- substring(annot$Array.Data.File,1,10) #adding annotation for Slide
annot$Array <- substring(annot$Array.Data.File,12,17) #adding annotation for Array
annot$Basename <- substr(annot$Array.Data.File,1,17) #adding annotation for Basename
annot <-annot[!duplicated(annot$Basename),] #remove duplicated row, due to Cy3 & Cy5 taking up separate rows
annot <- annot[!(annot$Characteristics.disease == "Crohn's disease"),] #remove rows with blood cells
methylset <- read.metharray.exp(targets = annot)

My .idat files are all in the same working directory as this R file is. But the error states it can't find my .idat files as follows:

Error in read.metharray(basenames = files, extended = extended, verbose = verbose, : The following specified files do not exist:201172580052_R02C_Grn.idat, 201172560046_R01C_Grn.idat, 201172560048_R02C_Grn.idat, 201172580052_R07C_Grn.idat,... (the error goes for all my .idat files)

Thank you in advance!

If you pass in a targets data.frame, then the expectation is that the 'Basename' column has the entire path to the directory that contains the .idat files. So you should be able to do

G.files <- paste0(annot$Basename, "_Grn.idat")
all(file.exists(G.files))

and have that return TRUE. Which it won't, based on your experience. So fix the annot$Basename column so it will do that correctly and you should be in business.