In Berkeley's idealism God holds the world together, enabling us to avoid chaos and experience a shared, orderly reality. But Schopenhauer is an idealist and an atheist. How can we then explain the shared experience of reality? What holds everything together? This seems quite odd.
The World Will, primordial, blind, and irrational, "holds the world together" in Schopenhauer's philosophy. It doesn't do a very good job of it, but then he was not called the father of pessimism for nothing. Schopenhauer's ethics was influenced by Oriental philosophy, in particular the Buddhist idea that the world is full of suffering, and the cause of suffering are human desires, confused, misguided, and ultimately futile. All living beings, and even inanimate objects, are merely "individuations" (his term) of the Will, the sole source of action in the world, and the root of its pointless suffering. Schopenhauer's ethics shares with Buddhists, and Stoics, the ideas of ascetic self-limitation and perseverance, the only way to "beat" the Will, which manifests itself in us through its incessant compulsive urges.
Schopenhauer's metaphysics, on the other hand, closely follows the classical German idealism, albeit mixed with also Oriental idea from the Upanishads: of Atman (soul) being mystically identical with Brahman (world). The world as Will for Schopenhauer is the Kantian thing-in-itself (and Kantian appearances form the world as representation). Unlike Kant, however, Schopenhauer admits a direct "intuition" of it, which manifests however not in artistic exaltation, as for Schelling, or moral law, as for Fichte, but in the base urges of the Will to Live. The dynamic of the Will can be described as the polar inversion of Hegel's boundless rationalism. His Absolute Geist starts from abstract notions, that are delimited through negations, and then sublated into the multiplicity of concreteness through negation of the negations. This dialectic is seen as both self-creation and self-cognition, at the end of which "the rational is the real". In parallel, the World Will tears itself apart, but not through rational moves of Hegel's Logic, but through violent urges that splinter it into a myriad of individuations, controlled by "umbilical cord" of unconscious volition.
Now concerning the title question. Idealism is the position that the fundamental "stuff" of the world is more like what we make contact with through our mental faculties of reason, understanding and intuition, rather than through our senses, as in the case of matter. Materialism is the opposite position. There is no logical connection between either of them and the existence of God as in the Western Christian tradition, an omnipotent, omniscient, personal being. There is no reason why the world, either ideal or material, can not "hold itself together", or be held together by some other mechanism or principle. That holding belief in God correlates with philosophical idealism is more a matter of tradition and psychology. Plato and Aristotle gave versions of idealism without such a God, but most later philosophers belonged to a Christian tradition, and so strived to find a place for him in their metaphysics. However Spinoza, and especially Hegel, reduced "God" to little more than a mere principle, which Schopenhauer inverted and termed the World Will.
Berkeley populated the world with entities, ideas, which were in their essence perceptions. Berkeley's famous formula was "esse est percipi", to be is to be perceived. Those perceptions, the ideas, are things which are necessarily perceived by someone, by some perceiver. There are, on the other hand, no substantial things behind the perceptions, in Berkeley's world. When I perceive e.g. a tree, there is never a real, substantial tree behind the perception, according to Berkeley. That is because, there are in Berkeley's world no entities that could exist without being perceived. This is the peculiar reason why God is required to "hold together" Berkeley's world. My perceptions of e.g. a tree are not grounded in a substantial entity, such as a real tree. Therefore the perceptions must be grounded in something else. And Berkeley concluded (after some more reasoning) that that something is God.
The situation is different with Schopenhauer, which is, in the aspect under discussion, following in the footsteps of Kant. In Kant's world, unlike that of Berkeley's, there are substantial things behind our perceptions. These are what Kant called the things-in-themselves. It is not only that things-in-themselves can exist without being perceived. They cannot possibly be perceived, as they are. When you perceive a tree, in Kant's world, there is something substantial behind the perception. This something is not a real material tree (which is why Kant has been called an idealist) but it is a real something, a thing-in-itself.
Since there are, then, substantial things behind our perceptions, these things, together with the perceiving subjects, are the grounds of our perceptions. Therefore Kant's world, unlike Berkeley's, is held together by these things-in-themselves (together with the perceiving subjects). Therefore the need for God does not arise in Kant's system, in the way that it arises in Berkeley's. And Schopenhauer's system, in the aspect under consideration, is just a variation on Kant's system. Schopenhauer upheld Kant's version of idealism, including the doctrine of the thing-in-itself. (Schopenhauer further held that there was just one thing-in-itself, which he called the Will.) That's why the need for God did not arise in Schopenhauer's idealism, as it did in Berkeley's.
Your question actually contains quite a lot.
- How can there be idealism without God?
- How can we explain the shared experience of reality?
- What holds everything together?
In any endeavor one should ask if there exist a causality linking these events together. Is the causality a necessary truth, or a contingent truth?
It is indeed a contingent truth. An ideology is basically a contingent truth, unique and relative to none but the beholder. Whether other people adopt it as their own, or choose their own ideology is the "sharing" in the experience.
Your last question seems to jump from everyday causality to a first cause such as in the idea of creation from nothing. The problem with this is causality itself is "held together" by time which is really a temporal expression of our imagination. Remove time from the equation and what is left is a beautiful world of objects and space, the two absolutes we can be certain about.
I don't see how there can be an idealism that has a basis without God, particularly anything in terms of an objective idealism.
The natural world doesn't know what oughtness is. Chemistry can't answer that. Physics can't answer that. Biology can only periferally answer that.
A scientific or rather scientistic approach would have us believe that we are "only chemistry, biology, and physics." The key word there is only. We are certainly chemistry, biology, and physics, but it would require a massive burden of proof to assert that that is all we are (eliminating other options systematically). Not the least of which is the actual human experiences of love, relationship, community, wonder, and awe. But beyond that the experience of freedom, deliberation, conscience, consciousness, imagination and other human experiences seriously call any reductive naturalist approach into serious question (and especially those that are only physicalist or materialist in nature).
A further problem is skepticism (ideological and philosophically as practiced), which in practice is undermining either all beliefs or all beliefs I disagree with. Both ultimately serve as the acids of belief. Without some basis of belief--its impossible to exist as a human being. Moreover the reductive approaches which are used tend to eliminate key aspects of reality (ie the big picture, history, and context). There is a certain populism or common sense involved in these approaches, but when you add specificity, nuance, history, context, or the big picture--these approaches usually falter. Not only that, their philosophical worldview as a whole is seriously brought into question by the above.
For instance, the illusion is that this kind of thinking will bring us freedom, but it actually leads to a very deterministic outlook--one in which human choice is entirely removed. Thats why Huxley undermined human dignity and human choice in his philosophy. That goes against the principles of freedom and human dignity that allow a platform like this to exist.
Moreover, such an approach runs aground of a kind of relativism. If we end in relativism, we end up with nothing. We have no notions of justice, accountability, or respect from systems of power. Governments, businesses, and the powerful become the rulers--because everything is relative. Its a Machiavellian outlook, rather than one in which freedom, dignity, justice, and everything we value most (for instance the US Constitution) is actually valued.