Do all the elementary particles in the standard model have their own wave functions? Is it possible to derive the wave functions of all particles in the standard model?
Would this be through the Schrödinger equation? 
 A: The seventeen particles in the Standard Model are the quanta of seventeen fundamental quantum fields: six for quarks, six for leptons, four for gauge bosons, and one for a scalar boson (the famous Higgs). Other than spacetime, which is where these fields “live”, today’s model of reality has only seventeen things to understand at a fundamental level! This is a remarkable triumph of reductionism, but physicists hope to someday do still better.
The Schrodinger equation describes only non-relativistic particles with mass. By contrast, quantum field theory, which is used in the Standard Model, can describe non-relativistic massive particles, relativistic massive particles, and (relativistic) massless particles. (There are no non-relativistic massless particles.)
So these quantum fields are capable of describing any kind of particle. However, the fields themselves are the primary underlying reality; their particle-like excitations are essentially secondary.
Relativistic quantum field theories don’t have wavefunctions. A wavefunction can describe where particles are likely to be, but it cannot describe particles being created and destroyed as happens at relativistic energies. The quantum fields of relativistic quantum field theory, by contrast, are fields of operators that create and destroy quanta.
For example, a single electron-positron field can describe, in theory, all the electrons and positrons in the universe, and allow their number to change, such as when an electron and a positron annihilate into two photons. This process is described as an interaction between the electron-positron field and the photon (i.e., electromagnetic) field.
Physicists are interested in the behavior of these fields even when they have no quantum excitations; i.e. a particle-free universe. This is the vacuum state, and it is nontrivial. For example, physicists do not understand what the energy density of the vacuum state is.
A: 
Is it possible to derive the wave functions of all particles in the standard model?

A wavefunction isn't a property that is unique for each particle type. The same type of particle can have different wavefunctions, and different particles can have the same wavefunction (or at least have the same form, depending on the system and what information you're keeping track of). You can also have wavefunctions for systems of particles, and they can be the same or different types of particles in the system. Therefore, there isn't a one-to-one correspondence between particle and wavefunction. 
This question is somewhat (not completely) analogous to asking if we can determine the position and momentum of all particles just depending on what type of particle. It depends on the system, the history of the particle, etc., but it doesn't depend on the specific particle. 

Would this be through the Schrödinger equation?

The Schrodinger equation just tells you how wavefunctions evolve over time (not taking into account relativity). It doesn't determine wavefunctions a priori. Once again, this depends on the system in question.
A: The underlying framework of nature is quantum mechanical, it is true. Wave functions are the solutions of quantum mechanical equations with or without boundary conditions, and are different depending on the conditions which include which elementary particles are  described.
The Schrodinger equations is not used for elementary particle interactions because it is non relativistic. The relativistic case is covered with different equations depending on the particle spins and other attributes. The Dirac equation is used for spin 1/2 particles, the Klein Gordon for bosons ( spin 1), and a quantized Maxwell's equation for photons.
When there are no potentials or interactions, the plane wave solutions of these equations are assumed to describe them. Quantum field theory was developed in order to solve scattering problems in particle physics , which are many body problems.
In field theory, a field is assumed to cover all spacetime for each particle in the particle table, an electron field a photon field, and these are the plane wave solutions of their corresponding quantum mechanical equations. On these plane waves creation and annihilation operators  describe the behavior of interacting particles. A type pf lorentz invariant ether. The interaction integrals are represented with Feynman diagrams which simplify the calculations.
So in a sense, the  particles are mathematically assigned a plane wave wavefunction, specific to their quantum numbers and mass, from the corresponding free (no potentials) equation,  but in describing a scattering for example, the wavefunctions are an underlying mathematical layer.
To describe the trajectory of a measured free particle, in a beam of electrons for example, one has to use the wave-packet concept, because plane waves give equal quantum mechanical probability for a particle to exist all over space-time.
