If an observer is needed to see something, but it is an observer that causes a quantum wave function to "collapse" into a classical state, how could we tell that the quantum wave function even existed in the first place?
Quantum mechanics, wave function and all that, was not invented out of thin air and imposed on nature. It was experimental results that could not be fitted with the classical theories of the end of the 19th century that created the need for a new mathematical theory/model to explain observations.
With classical mechanics we can solve the newtonian gravitational equations and get a mathematical formula that fits the orbits of the planets around the sun. We can even add the corrections of General relativity and get even more accurate results. This is described by a very complicated function, written in planetarium computer programs for example. This mathematical function exists and describes the orbits giving (x,y,z) at time t for a planet.
Does the planet know about this? i.e when found at x,y,z as predicted the function collapses?. Ridiculous, no? It is just a measurement.
Quantum mechanics differs from classical mechanics as the function that gives the "orbits" of the electrons around the nucleus is a probability distribution for where to find the electron , x,y,z at time t. This probability distribution is a hard mathematical function as good as the planetary orbit describing functions of classical mechanics, written on paper and in computers. A measurement of the electron's position gives one point in the probability distribution , cumulatively giving the atomic orbitals.
If an observer is needed to see something,
a measurement is needed both for classical orbits and quantum mechanical orbitals.
but it is an observer that causes a quantum wave function to "collapse" into a classical state,
If by "classical state" you mean a number in our counters, an observer is needed either in the telescope looking at the moon's location or at the value of the electron's position in recording the orbitals, point by point.
how could we tell that the quantum wave function even existed in the first place?
Well, the functions exist written on paper or in the computer when complicated, and they fit the data, whether classical (moon orbit) or quantum mechanical , the orbitals.
Mathematical functions are not balloons that burst, they have been fitted to measurements and are a tool to predict new measurements, and when they do they are successful and developed further .
Let me first correct you on a point that you make: Wave function collapses into an eigenstate of the measurement operator. The term "classical state" is not standard terminology.
Now let me try to answer your question. It seems like you're asking whether there exists a physical reality separate from our observation (measurement). The answer is that we do not know, it is an assumption that we (usually) make.
Side note: Additionally making the assumption of locality and free will introduces an upper bound on the amount of correlation two systems can have. Quantum mechanics violates this upper bound, meaning that at least one of the assumptions must be incorrect. See Bell's theorem if you're interested.