When an electron becomes "excited", where is the energy "stored", and as what?

There are multiple contributions to the energy of an electron in an orbital. Let us consider atomic orbitals for now, and then think about extending to molecular orbitals.

Atomic orbitals are defined by the quantum numbers, the first 2 of which are the major factors in energy:

The principle quantum number (n) can be thought of as a pure electrostatic term, signifying the average distance between the electron and the nucleus. The farther apart they are the more electrostatic potential energy is stored (just like gravitational potential energy).

The secondary quantum number (l) signifies the angular momentum of the electron, you can think of this as internal kinetic energy (classically, how fast the electron is moving around the nucleus).

The other two quantum numbers do contribute to the energy, but in more complex ways (especially relevant in systems with multiple interacting electrons).

Molecular orbitals are much more complex, and harder to wrap one's head around. But we can borrow tricks from matrix algebra, and show that molecular orbitals are linear combinations of atomic orbitals of each of the constituent atoms, and because of various conservations laws, we can also add up the energies and angular momenta of each of the constituent orbitals, and know that the rules and trends are the same.