This is a similar question to the one about the lack of use of usual topologies in measure theory. By usual topology here is meant the Hausdorff-Kuratowski-Bourbaki concept, based on open sets, or alternatively, closed sets. In ordered spaces a variety of convergences are defined. Hoever, in their general cases they do not correspond to convergence in any topology. The question is : is there some other concept of topology so that appropriate instances of it may give precisely one or another set of convergent sequences in order spaces ?
Pretopological spaces generalize both the topological notion of convergence and most kinds of order convergence. In a toplogical space X, one can define a satisfactory notion of convergence by filters. A filter in X is a nonempty family of subsets of X that is closed under finite intersections, supersets and that does not contain the empty set. The systems of neighborhoods (not just open neighborhoods) of a point forms a filter. A filter is said to converge to a point if it contains all neighborhoods.
One can abstract from this concept and define a pretopological space by a set X of points and for every point x in X a Filter representing abstract Neighborhoods such that x is an element of every neighborhood. A filter now converges again to a point if it includes every neighborhood.
There are various criteria by which one can identify pretopological spaces that are actually topological spaces. The Handbook of Analysis and its Foundations by Eric Schechter contains a lot of material on various forms of convergences and is a good reference for such questions. But there is no reason that the various noions of order convergence are topological. Actually, using the Bourbaki classifcation topological structures and order structures are separate types of structures. They only coincide if one wants by, say, using the order topology on a completely preordered set, in which case topological convergence also leads to a nice type of order convergence.