The final chapters (11-13) of Clouser's The Myth of Religious Neutrality (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) present a brief overview of Dooyeweerd's philosophy, applying it in particular to society and the state. I'll try to give the gist of it here.
This "law framework theory" starts off by recognizing that reality as we experience it has many different aspects, each with its own properties and laws. Then the theory asserts that these aspects cannot be reduced to each other (so psychology is not "nothing but" physics and biology, for example). Instead, each aspect of reality is directly created by God, and depends directly on him for its existence. Here's a provisional list, which is probably best read from bottom to top:
fiduciary
[love-]ethical
justitial
aesthetic
economic
social
linguistic
historical [or cultural or technological]
logical
sensory
biotic
physical
kinetic
spatial
quantitative (p.244)
Next the theory talks about "the natures of things" (p.260). Every thing or entity that exists functions in all of the aspects, whether actively or passively, and is subject to the laws of each aspect. So a rock functions actively in the quantitative, spatial, kinetic and physical aspects, but also functions passively in the other aspects. For example, it might be a stone in an animal's den (biotic), or used in a religious ritual (fiduciary). As such, all entities are described by type laws, defined as "laws which range across aspects determining which properties of different aspects can combine in individuals, and thus determine the types of individuals that are possible" (p.268).
In particular, each type of thing is characterised by one of the aspects (its leading aspect or qualifying aspect), which governs the properties of the thing as a whole.
This is getting a bit abstract, so let's apply it to society.
There are many different communities in society, such as businesses (characterised by the economic aspect), families (characterised by the ethical aspect), religious communities (fiducial), orchestras (aesthetic) and the state (justitial). These communities cannot be reduced to each other, because they have fundamentally different purposes, as described by their leading aspect. Each community has authority to act within that aspect (as in Abraham Kuyper's principle of "sphere sovereignty"), and this authority is derived from God and is subject to the laws God has created for that aspect. So "on this view there is no institution which can rightfully claim to have supreme authority for the whole of human life" (p.291).
So what is the state? It is not the ultimate authority, not even when that authority is given by the majority vote or is restricted by some set of individual rights that the state cannot violate.
By contrast, the law framework theory sees the state as the bearer, not the creator, of the authority it wields in enforcing justice. The will of the majority decides who shall be the bearers of that authority, but the authority itself derives from the law framework of creation and thus, ultimately, from God (p.311).
And what is the state for? It is not for regulating every detail of our lives. Instead,
The state ... has its own distinct kind of authority, an authority qualified by justice — more specifically, public justice. Its ability to carry out justice must extend to the whole of the public within its territory, of course. Nevertheless, its authority is limited to but one aspect of that public. And let me emphasize that it is precisely because justice is an aspect of all individuals and communities, that the state need not subsume them all as its parts in order to exercise its proper authority with respect to them. In other words, state authority need not be elevated above all others on the excuse that it needs totalitarian authority to ensure justice to all individuals and communities (p.296).
It is this kind of approach to reality, based on belief in a God who created the many different aspects of reality that we experience, that can give the state a clear role in wider society, without dominating society, and without being weak and ineffective in operating within its sphere of influence.