Probably not buffalo- Laura mentions that they are long gone, and she wishes she could see one when they visit the old buffalo wallows.
I've camped a lot, so I get the idea of going the bathroom outside- on their travels, that makes sense to me. I'm just really more interested in permanent settlement and what kind of sanitation was set up. It appears they move the outhouse whenever the pit gets full. I'm just wondering if the praire towns were "shit all over the street" type places that Medieval cities seem like they must have been (when it was tossed into the street to get it out of the house!), or if it was actually discarded in a reasonably clean manner.
ROFL, no, and neither were Medieval cities. Not nearly like pop-history likes to show. (And if you go back to ancient times, some places had functional public lavatories, and in Rome launderers put out pots for people to piss in so they could collect the urine and use it in cloth-cleaning.) Outhouses WERE moved when they got a bit filled, but that takes much longer than you'd think as they're quite deep and they do compost. But if one started really being past its use, they'd backfill and dig a new one. Backfilled outhouses are actually great for archaeologists--not only do people drop stuff, they deliberately use them as garbage dumps and you'll find broken crockery and glass, buttons, bones and basically anything with enough substance to last longer than the moisture does. (No, it doesn't stink.) Until the late 19th century and the start of indoor WCs, everywhere, in town, out east, and out west, would have privvys out back. Chamber pots were really only for when it's too dark or cold at night to go out (or if you're out on the frontier when it's legitimately dangerous to walk twenty yards out behind the house in the pitch black.) And they'd be emptied down the privvy.
And yes, there'd still have been buffalo chips dried out, and then OLD (but never new) cattle droppings. The buffalo were only 'long' gone if you're an eight-year-old kid with no concept of time (plus dried dung lasts a long time.) The actual dates the Ingallses were on the prairie, the great mass millions of buffalo were drastically reduced, but they were hardly extinct. Horse poop works, too, though there weren't nearly as many (there were never large populations of "wild" horses in North America, at least not after the original ancestors of the horse went extinct thousands of years earlier. The natives had a few from the Spanish that were small feral populations, but "wild" horses were and still are just feral offspring of released cavalry remounts and farm horses, so there were never nearly as many as cows or bison.) And the older and drier, the less painfully stinky burning it would be.