What exactly gives electrons a charge? I understand how in molecules, an imbalance between electrons and protons give ions charges and I also understand that there is really no positive or negative charge, they are just names assigned to opposite charges, but I am just very unsatisfied with not actually knowing what an electron is and why it has a charge.
I know electrons have a negative charge and that they are subatomic particles made up of even smaller particles,
They are not, so far as we know, made up of even smaller particles. It behaves like a particle that is not composite and is basically a zero radius point in space called a point particle, to the fullest extent that it is possible to test this experimentally. As explained in the point particle link:
[T]here is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket can be represented as a quantum superposition of quantum states wherein the particle is exactly localized. Moreover, the interactions of the particle can be represented as a superposition of interactions of individual states which are localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.
For example, for the electron, experimental evidence shows that the size of an electron is less than 10^−18 m. This is consistent with the expected value of exactly zero.
So far, the fundamental particles we know about are six kinds of quarks, three kinds of charged leptons (including the electron), three kinds of neutrinos, the W+ boson, the antiparticles of all of these particles, the Z boson, the photon, eight kinds of gluons, and the Higgs boson (each kind of quark comes in three colors and each of those can have left or right parity, each kind of charged lepton can have left or right parity, all neutrinos in the Standard Model are left parity and all anti-neutrinos in the Standard Model are right parity). There is also one hypothetical particle, the graviton, which a great many scientists (but not all) believe is an additional fundamental particle.
This is reality as we observe it, and the Standard Model does not provide any deeper explanation for it. Many extensions of the Standard Model, such as supersymmetry, propose that even more fundamental particles exist. But, science has not pierced successfully yet to a layer more fundamental than the Standard Model.
I am just very unsatisfied with not actually knowing what an electron is and why it has a charge.
So are lots of scientists. But, they haven't come up with any better explanations. At best, many theoretical physicists would suggest that it might be related to M-theory (i.e. string theory) somehow or other. But, there is no realized, specific model implementing string theory that answers these questions in any meaningful way.