Was the understanding of QM fundamental to the creation of transistors and silicon semiconductors?

A short timeline of the development of the transistor and Bardeen's role in it.

From John Bardeen (Wikipedia)

The assignment of the group was to seek a solid-state alternative to fragile glass vacuum tube amplifiers. Their first attempts were based on Shockley's ideas about using an external electrical field on a semiconductor to affect its conductivity. These experiments mysteriously failed every time in all sorts of configurations and materials. The group was at a standstill until Bardeen suggested a theory that invoked surface states that prevented the field from penetrating the semiconductor.

Bardeen suggested a theory that invoked surface states that prevented the field from penetrating the semiconductor.

As stated by Bloch's theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves [2]

$${ {\begin{aligned}\Psi _{n{\boldsymbol {k}}}&=\mathrm {e} ^{i{\boldsymbol {k}}\cdot {\boldsymbol {r}}}u_{n{\boldsymbol {k}}}({\boldsymbol {r}}).\end{aligned}}}$$

Here ${u_{n{\boldsymbol {k}}}({\boldsymbol {r}})}$ is a function with the same periodicity as the crystal, $n$ is the band index and $k$ is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions  The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected.

So to understand surface states involved knowledge regarding the Schrödinger Equation and Bloch's theorem, both based around Q.M.

In 1957, John Bardeen, in collaboration with Leon Cooper and his doctoral student John Robert Schrieffer, proposed the standard theory of superconductivity known as the BCS theory (named for their initials).

Again, quantum concepts are involved here.

So although I cannot say for sure how much Q.M. was involved in the invention of the transistor, or at what stage in his life did Bardeen first apply its concepts, there is certainly circumstantial evidence that Q.M. contributed to the development of the transistor.

You could also look through the published papers of the people involved in transistor development to see how much Q.M. was incorporated in them.

The explanation of the electron distribution in a semiconductor can only be explained by means of QM, as classical theory can't prove the properties of transistors. Without going in much detail, this can be explained by QM as how the wave of electron behaves in a system with periodic structure (such as semiconductors) giving rise to "forbidden" energy levels, called gaps. These gaps are crucial to prove how transistors work.

On a side note, transistors were discovered much before (~10 years) this band-gap theory was developed, and it was not until the reason was totally understood that they begin using them for applications such as computers.

To explain the physical working of a (bipolar) transistor, quantum mechanics is essential in the following sense. You need to know that electrons in a semiconductor occupy allowed energy bands and that there is a so called conduction band where electrons can move freely and an almost fully occupied valence band where missing electrons can be considered to behave as positive quasi-particles called holes. The occupation statistics is Fermi-Dirac statistics which includes the Pauli principle. Furthermore, by introducing impurities with quantum mechanical ionization energies that donate or accept electrons you can dope the semiconductor, i.e., you can create regions of the semiconductor with predominantly electron or hole conduction, which is essential for the device operation. With these fundamental results of the quantum theory of semiconductors, you can describe the working of transistors pretty much classically. For example, electrons and holes can be considered to be point particles which can be generated or recombine with a certain probability. Further, they can diffuse and drift in an electric field experiencing phenomenologically a "resistive friction" in the crystal lattice.