Do electrons have inertia? I don't know quantum mechanics and know a little bit about mechanics and a very little bit about subatomic particles but I am just a curious to know the answer to my question. My question is if 'inertia' is unwillingness to of an object to change its state of rest or uniform motion then can I say that electrons have no inertia as they are continuously moving by themselves without the need for an external force to move them and therefore have no mass since inertia is due to mass?
 A: In physics, the phenomenon of inertia is in a category where an exhaustive explanation for it is not available (and may never be). The point is: in order to have a theory of physics at all the properties of inertia must be granted. So that is what physicists do; the properties of inertia are granted.

The closest thing to inertia that we have some understanding of is the phenomenon of inductance.
In the physics of currents and electromagnetic fields: a changing magnetic field will induce a current in an electric circuit. Conversely, change of current strength generates a magnetic field. This effect is stronger when the current-carrying wire is coiled. The more windings the coil has, the more accumulation of effect.
Imagine an electric circuit, cooled to a low enough temperature so that it is superconducting. So: once current is started that current will keep going round without loss. Let that circuit also include a coil with strong self-inductance. The self-inductance means that the coil will oppose change of current strength.
I use the word 'opposition' here, instead of 'resistance', because the effect is fundamentally different from resistance. Resistance is like drag force. A resistor resists flow of current. When current flows through a resistor, some of the energy of the current is transformed to thermal energy. The larger the current the larger the generation of heat.
A superconducting coil with self-induction will not resist current. But if you apply an electromotive force, to either increase or decrease the current strength, then due to the self-inductance the current strength will not jump up/down instantaneously; the current strength will have a rate of climb/descent. The stronger the self-inductance, the more electromotive force is required for the same rate of change of current strength.

Historically the phenomenon of inertia has often been described as something innate to objects. However, that is logically untenable.
Here is a thought demonstration:
Take the case of you using your physical strength to push a rod into a lump of clay. In order to push, you need leverage. You have to brace yourself some way or another. Now visualize a marble that you shoot into that lump of clay. The marble penetrates the clay in proportion to its velocity. It is tempting to think that the marble pushed itself into the clay, but that idea doesn't make sense. (There is a Baron von Munchhausen story where the baron, while horse-riding, gets stuck in a swamp. According to the story the Baron grabbed his own hair and managed to pull himself out of the swamp. If you think the Baron von Munchhausen story is unphysical, you must also come to the conclusion that the idea of the marble pushing itself into the clay makes no sense.) Something is giving that marble leverage, such that a force is required for the marble to undergo a change of velocity.
We have that inertia opposes change of velocity, but at the same time inertia is entirely transparent to uniform velocity.


We have that inertia couples to inertial mass (which of course is why it's called 'inertial mass').
The introduction of special relatvity showed that the concept of inertia has to be widened. The article that introduced special relativity, 'On the electrodynamics of moving bodies,' was published in 1905. In that same year Einstein used a thought demonstration to point out an implication of special relativity. The thought demonstration was a box, with perfect internal reflection, with light in it bouncing from wall to wall. When you push that box to change its velocity, then you have to take the dynamics of the bounces of the light into account. Einstein proceeded to show that the the presence of the bouncing light inside the box gives rise to an additional amount of inertia, in propotion to the energy content inside the box. The reasoning of that thought demonstration generalizes: in terms of special relativity: when you have a definable energy in a volume of space, there is a corresponding opposition to change of velocity of that amount of energy.
Put succinctly: the energy of a bound system has a corresponding inertial mass.
(To be clear: this is not about explanation of inertia; this is about recognizing that in the light of special relativity the scope of the concept of inertia has to be widened.)

To your question specifically:
It's not clear to me what you refer to with the statement: "electrons are continuously moving by themselves". Electrons do have inertial mass. If you create a beam of electrons then a force is required to make that beam bend.
A: 
Inertia is the resistance of any physical object to a change in its velocity. This includes changes to the object's speed, or direction of motion. An aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed when no forces act upon them.

You state:

if 'inertia' is unwillingness to of an object to change its state of rest or uniform motion

Not "unwillingness", but "resistance", and this can be measured in experiments and codified in the law F=ma , Newton's second law, where m is the mass measured when accelerated.

then can I say that electrons have no inertia

But they do as they are particles with mass which can be measured by applying acceleration.

as they are continuously moving by themselves without the need for an external force

In Newton's first law , everything that is in motion continues in motion, which is true for electrons too.

Law 1. A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.

Are you thinking of electrons bound in atoms ? That is an entirely different story, not classical mechanics and its inertia, that needs quantum mechanics to understand, which you should study if you really want to understand particle physics.
A: Despite quantum effects an electron without external forces will act as any other macroscopic object without external forces. In particular it will not accelerate by itself and it has constant inertia.
