Which fundamental force underlies inertia? We learn about the four fundamental forces (gravity, electromagnetism, strong, weak). I am curious about how these four forces mediate/cause/drive everyday phenomena. In particular, it has never been clear how inertia arises as a result or is caused by these forces.
So taking a car accelerating while driving up a hill. The fundamental forces will act in numerous ways to require energy expenditure: 
(1) The car moves uphill against earth's gravitational field 
(2) Electrostatic interactions cause friction with road surface
(3) Air are jostled/displaced (electromagnetic repulsion between molecules of car and air)
But the increasing inertia of the car is mysterious. If the four forces underlie all action/reaction, which one of them is responsible for requiring the energy to accelerate the car and increase its KE? 
Among household names, it seems that Einstein (and perhaps Feynman) supposed that inertia was related to gravity in a "Mach"-ian manner. My reading comes up blank when I try to find a consensus view. It seems inertia is taken for granted as a fundamental aspect of reality but is not explained/explainable in terms of the four fundamental forces. Thoughts?
 A: Inertia, or the Law of Inertia is generally used by physicists as another name for Newton's First Law. Inertia is not something that can be possessed by an object.
In other words, inertia describes the tendency of objects to either remain at rest or continue moving at a constant velocity unless the object is acted upon by some external force.* There is no physical property of the object called inertia, and as such, inertia cannot increase or decrease. It is best understood as a principle of motion. (Personally, I usually encourage students to simply not use the term, as the physics meaning of "inertia" does not align well with the vernacular use, where it means "sluggishness" or "resistance to motion.")
To answer your question, inertia does not arise from forces, nor is it a result thereof. Indeed, it "is not explained/explainable in terms of the four fundamental forces." It is instead a law describing what happens in the absence of net force -- it helps explain what the fundamental forces do to objects.
So why does it take more energy to move the car uphill than it does to move the car on a level plane? This can be understood through Newton's Second Law:
$$
\sum \vec{F} = m \vec{a}.
$$
If you want the car to move at a constant velocity, then $\vec{a} = 0$, so 
$$
\sum \vec{F} = 0.
$$
We can break this up into the $x$ (horizontal) and $y$ (vertical) components.
On a flat surface
$$
\sum F_x = F_{engine} - F_{friction} = 0
$$
and
$$
\sum F_y = F_{normal}-F_{gravity}=0
$$
where $F_{normal}$ is the force of the ground pushing up on the car's wheels. (I omitted air resistance for now, but if you are bothered by that feel free to think of the friction term as including air resistance as well).
As you can see, in order to satisfy the first equation, $F_{engine} = F_{friction}$.
However, in the inclined case, where the car is traveling up hill, the engine will have to overcome not only the force of friction, but also the force due to gravity, which will act in part to push the car back down the hill. Thus, the engine must work harder in order to keep the car moving at a constant speed (zero acceleration).

*Note that this technically only applies in what is known as an inertial reference frame, although if you haven't learned about those yet, you may ignore this for now.
A: This answer is meant to supplement Bunji's.
As Bunji stated, the principle of inertia is aptly constructed in Newton's First Law. For most, this is usually sufficient.
But you seem to be asking a much deeper question: 

it has never been clear how inertia arises

That's because it's not a clear subject - the origin of inertia is a deeply unsolved problem. As it is, science is a way of turning metaphysics into physics - that is, we begin to speculate about things when we lack empirical knowledge about them, and that involves philosophy. How inertia arises in nature has a rich philosophical dialogue. For instance, the stanford encyclopedia is a great place to start (and find further references). Special relativity (SR) is how Einstein made the Galilean principle of relativity compatible with electrodynamics (Maxwell's Laws). In this context, inertia is the equivalence class of "inertial frames of reference," which are all in uniform relative motion (As Bunji said, $F_{net} = 0$ for each) and in which "the equations of mechanics hold good," as Einstein stated. But this will leave you unsatisfied, as it did Einstein, since gravity clearly has a role to play with inertia. 

Among household names, it seems that Einstein (and perhaps Feynman) supposed that inertia was related to gravity in a "Mach"-ian manner. My reading comes up blank when I try to find a consensus view. It seems inertia is taken for granted as a fundamental aspect of reality but is not explained/explainable in terms of the four fundamental forces. Thoughts?

Indeed, Einstein was inspired by Mach's philosophy (and his critics) in discovering his theory of gravity - General Relativity (GR). GR is not fully Machian, however it is background independent in the sense that the "the fundamental properties of the elementary entities consist entirely in relationships between those elementary entities (R2 of here)." This makes it incredibly unique from every theory that preceded it - including SR. Experimentally, GR has been shown to be very accurate and predictive. 
Now, GR does not provide an origin for inertia in the way you seem to expect. Rather what it does do, and it took years for Einstein to come to terms with this, is unifies gravity and inertia as two different manifestations of the same thing - the inertio-gravitational field (the metric). The unification is seen in recognizing the equation of motion for free-fall particles (the geodesic equation) in GR as the equation of inertial motion. It is analogous to how Maxwell unified electricity and magnetism into electromagnetism. Thus, in GR the principle of inertia is generalized by the geodesic equation. As the author of this states, 

The unification of these two fields into one inertio-gravitational field that splits differently into inertial and gravitational components in different coordinate systems is one of Einstein's central achievements with general relativity.

After doing this, Einstein became enthralled with unification schemes which left a legacy that is still active today. 
As Feynman explains (or perhaps the look he gives says it all), the "why" questions are very difficult to answer, and "why does inertia exist" is a particularly difficult one to answer. 
