The error is from misplaced parentheses, so these minor variations on your code will work.
L[grep(paste(id1,id2,sep=""), L)]
# character(0)
L[grep(paste(id2,id1,sep=""), L)]
# [1] "catdog"
Alternatively this is a regex one-liner:
L[grep(paste0(id2, id1, "|", id1, id2), L)]
# [1] "catdog"
That and some patterns in the comments will also match dogcatt. To avoid this you could use ^ and $ like so:
x <- c("dogcat", "foo", "catdog", "ddogcatt")
x[grep(paste0("^", id2, id1, "|", id1, id2, "$"), x)]
# [1] "dogcat" "catdog"