We know pretty early in this movie that we're going to be dealing with a third-person narrator. Yes, there are some characters that turn to the camera and speak to us directly, which might make us feel like they're the first person narrator. But so many characters do this that it's clear that the only person really in control of this movie is Director Mel Brooks (who by the way likes to insert himself whenever he can into the action).
By the end of the movie, we realize that we must be dealing with a third-person narrator because even the boundaries of the movie itself end up disintegrating. When the people of Rock Ridge ride into their fake replica town to finish off the bad guys, the fight eventually breaks out of the movie set for Blazing Saddles and ends up spilling onto other movie sets, causing mayhem throughout the Warner Brothers studio lot.
The only type of narrator who could handle this sort of thing is a third-person omniscient, although we can honestly say that even an omniscient narrator might have trouble making sense of everything that happens in this movie… or stifling his giggles, for that matter.