The buckboard might in fact be the name given to the flat board that comprised the carriage (on which the seat was placed) as well as being the name given to the type of carriage itself. So the mud was up to the base of the board, in other words.
It's an unusual passage. There are a couple of strange metaphors. Firstly, we're told he was riding as fast as the wind (keeping pace with the wind), but since the horses were running through deep snow, it's difficult to see how that could be so. Secondly, the horses were trudging through snow but, immediately behind them, the carriage they were pulling was deep in mud. How come?
A mention about the "sleigh rails" - this conjures up an image of an alternative type of carriage, the type of sleigh, with bells, that in many cultures you would expect to see Father Christmas' reindeers pulling (the sleigh rails being what that type of vehicle runs on in the snow). It's hard to know, in that case, exactly what the author was describing - although one possibility, given that buckboard is broadly speaking an American term, is that the location is the American Mid-West where, in winter, settlers would certainly have had to cope with heavy snow. That's purely some speculation on my part.