Jerry, nice Mossberg! The Model S51M is a good little rifle. I have the Model M152 - same action, except it has the detachable clip magazine.
You are missing the No.4D aperture for your No.4 peep sight. You'll notice the threaded hole in the windage adjustment bar. You can still use it like a "ghost ring" sight. This sight also swings over to the side, so you can install a 3/4" tube scope without removing the peep sight. Just loosen the big lower knob, swing the sight over and lock it down with the knob. It won't clear a modern 1" tube scope though. Is the rifle grooved for a scope, or does it have the two big threaded holes on the left side of the receiver, about 2.5" apart? If the latter, Havlin has a scope mount that'll work good with modern scopes. The original mounts can be pricey - if you can find one.
The "S" prefix denotes Mossberg rifles made for and marketed by the Speigel mail-order chain many years back.
Here is a great Mossy reference website:
http://www.geocities.com/sloopitol2/1Openingpage.html
The page for your rifle:
http://www.geocities.com/sloopitol2/modela51m.html
For parts, try Vic Havlin, great guy and he literally "wrote the book" on the Mossy twentytwo rifles.
http://www.havlinsales.com/
He may have the sight aperture, but it's scarce for that particular sight - I know, I'm looking for one too. You can bolt a later S130 peep sight to this rifle, the screwholes match. Also, Lyman's Model 57 with the 57MS base will bolt right on, as will the very similar Mossy S330. Havlin usually has the S130 in stock, about $65. Also, check on Auction Arms and Gunbroker, even eBay - search on "sight" and wade thru the stuff til you find what you're needing.
If you really want a good peep sight, and can't find one online, I have a complete S130 I might be persuaded to part with.

That's a cool little front sight, too. I prefer the skinny tapered post with the bead - it seems to give me the best accuracy for most targets. The ring aperture can be good, for certain targets. Diameter and distance is critical though.
Just please, dont ever cut down the Mannlicher style stock - these are getting scarcer! The front wooden piece and the metal between them can be removed, but is not advised. Collector value goes down the tubes.
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Sorry so long winded - did I mention I really like old Mossbergs? I have five of them right now.

Again, great find!!