Whether or not observation is the only way in which a wave function can collapse depends on what you mean by “collapse”, and that word is used by various people in reference to different aspects of the measurement and observation process – which can be considered as happening in two stages.

The setting involves a system in a pure quantum state which may have been prepared as an eigenstate of some observable (such as spin relative to a particular direction), and so is a nontrivial superposition of eigenstates of some other observable (such as spin relative to a different axis) which we now want to measure.

In the first stage, the system of interest interacts with a larger more complex system which is not fully known and so is in a statistical mixture of pure states (represented by a density matrix rather than a single state vector). If the larger system is suitably designed as a measuring apparatus, then the interaction leads to the state of the combined system approaching a statistical mixture of states in which the subsystem of interest is in an eigenstate of the observable and the measurement apparatus is in a related state which involves some macroscopic feature (such as a pointer, a readout panel, or a bright spot on a phosphor screen) which has a corresponding humanly visible value. Henceforth the system acts as if it is in just one eigenstate which is not yet known but is subject to classical probabilities. This process eliminates the possibility of future interference between the eigenstates that was possible while the state of the system was in a pure state (represented by a coherent wave function) and so is often called “decoherence”; and since it reduces the system to being effectively in just one eigenstate it is often identified with “collapse of the wave function”. It actually happens in almost any interaction with a complex system (even when there is no humanly visible related macroscopic property of the system). So, for those who identify decoherence as collapse, it is indeed possible for collapse to occur without observation.

But after this kind of “collapse” we still don’t know what the measured value actually is, even though we can think of it as having just one of several precise values – each with some known probability.

The second stage of the observation process is where the conscious observer notices which value is present. Some people think of this as where the “collapse” happens, but here it is not really collapse of the wave function but rather of the classical probability distribution (similar to the case of a coin toss which starts of in a stochastically mixed state and collapses to just one case when we see the result).

The difference from a coin toss is that in that case we assume that all along the system was really in whatever particular state we eventually observe, and that state could have been determined with certainty just by making more observations at the start; whereas in the quantum situation the uncertainty seems to be essential until we actually experience the result. This leads to a philosophical problem for those who think that the quantum state is a property of the system itself rather than of its relation to the observer as it seems to imply that the experience of a conscious observer has some physical effect on the universe and raises the problem of Wigner’s friend who watches an experiment before Wigner does and seems to collapse the wave function even though the friend is himself just a complex quantum system who Wigner sees with a wave function that does not collapse until the information reaches his (Wigner’s) own mind.

To my mind this is resolved by seeing the quantum state as a description not of the universe but of its relationship to the observer; and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI).