R, 68 66 bytes
for(i in 1:100)write(max(i,paste0("Fizz"[!i%%3],"Buzz"[!i%%5])),1)
for(i in 1:100)write(max(i,paste0("Fizz"[!i%%3],"Buzz"[!i%%5])),1)
I must be missing something here because this is way too simple. I know that you really shouldn't use loops in R but like this is a bit smaller than the first R one I saw so yeah.
Edited about a day after the original: you can get rid of the curly braces to save 2 characters.