Let me record the answer from the comments here.
Suppose $(W,S)$ is a finite Coxeter system, with associated root system $\Phi$ in Euclidean vector space $V$. (I don't think the assumption of crystallography is needed in anything I'm about to say, but if you care specifically about Type D, that is a crystallographic type so things will be even better there.)
The Coxeter arrangement associated to $\Phi$ is the central hyperplane arrangement with hyperplanes orthogonal to the roots. As mentioned in the question, the Coxeter complex $\Sigma$ of $W$ can be realized as the order complex of the lattice of intersections of the Coxeter arrangement of $\Phi$. This is a flag simplicial complex, homeomorphic to a sphere. Its facets correspond to the elements of $W$. A nice reference for this construction is Section 11.6 of T. Kyle Petersen's "Eulerian Numbers" textbook.
EDIT ADDED MUCH LATER: Sorry, the previous paragraph was wrong; in particular, the Coxeter complex is not always an order complex of a poset. The Coxeter complex is obtained from the Coxeter arrangement by intersecting the arrangement with a sphere. It is true in Types A and B that the Coxeter arrangement is an order complex of a poset (in Type A that poset is the Boolean lattice, as was mentioned by the question-asker), but e.g. in Type D the Coxeter complex is not the order complex of any poset. It is true that the Coxeter complex is always a flag simplicial complex.
For finite ("spherical") Coxeter groups, the Coxeter complex as mentioned is homeomorphic to a sphere; but even better, it is polytopal. The permutohedron of $W$ (sometimes called "Coxterhedron" in earlier references) is the convex hull of the $W$ orbit of any point in $V$ not on any hyperplane in the Coxeter arrangement. Its combinatorial structure does not depend on which point in the fundamental chamber we choose. The $W$-permutohedron is a simple polytope, and its vertices are in bijection with the elements of $W$.
In fact, there is another way to realize the $W$-permutohedron as a zonotope: it is the Minkowksi sum $\sum_{\alpha \in \Phi^+}[-\alpha/2,\alpha/2]$ (here the notation $[-\alpha/2,\alpha/2]$ means the line segment joining $-\alpha/2$ to $\alpha/2$; you can just think of this as the Minkowski sum of the roots of $\Phi$).
The permutohedron and the Coxeter complex are basically the same thing: precisely, the Coxeter complex is the dual of the permutohedron (the dual of a simple polytope is a simplicial polytope). See for instance the article "Coxeter-associahedra" of Reiner and Ziegler mentioned above for a quick review of these facts.
This polytopal construction gives a rather concrete model for the Coxeter complex in all the finite types. You can work out the combinatorics of the Type D case specifically from this construction, but there are also probably places where it is spelled out already.