Does the Doppler effect happen instantly?

The question is, in antenna's frame after acceleration, (which is indeed inertia) how can the frequency of the light still be shifted?

The light frequency was always shifted in this frame. (i.e. the frequency was always different in this frame than in the frame where the receiver is intially not moving)

You have two inertial frames, we can call them "A" and "B". Initially the receiver is at rest in frame A, and then it accelerates until it is at rest in frame B. But frame B wasn't created by this action. Frame B always existed. The only difference is that initially the receiver is at rest in frame A, so it measures the source frequency in frame A. Later the receiver is at rest in frame B, so it measures the source frequency in frame B.

But the frequency of the light in the frame B never changed, so there's no reason to worry about how it could have changed instantaneously.

Relative speed exists only between source of light and observer. To compute doppler shift you need to know this relative speed. If you discard the source of light, then to what you assign a velocity?

No, the relevant speed for doppler shift is not the current speed between source and observer. The light emitted by the star 100 years ago exists independent of the star after being emitted. The doppler shifted light you see is the light that fills the space between you and the star. Only you and the light in the local space around you is important, thus the distance to the star or what happened to the star in those 100 years is irrelevant. The star might not even exist anymore. The relevant speed is the relative speed between your frame before the acceleration and your frame after acceleration. The doppler shift depends on how much you accelerated. Acceleration is absolute not relative.

After all, the light transmitter is far like 100 light years away, so when antenna moves, in antenna's frame we would expect some kind of delay to see a movement for the the far star (transmitter) and ofcourse the Doppler shift effect (antenna is at the rest in its frame, so the only reason for receiving doppler shifted frequency would be the movement of transmitter itself in this frame). But there is no such delay in formulas at least.

No we would only expect a delay if the star is the one accelerating. If we are accelerating with our antenna there is no delay, because we accelerate.

Maybe an analogy can help you:

Imagine a cloud at rest above your head. You are initially standing at rest at the ground. The cloud is emitting raindrops and each raindrop is falling for a time T until it hits the ground. Initially the raindrops will fall vertically on your head. However if you start moving the drops will hit you in the face at an angle. From the angle at which the drops hit you, you can calculate how fast the clouds are moving relative to you (assuming the clouds never accelerate). In this analogy you can only observe the clouds indirectly by measureing the rainsdrops hitting you. If you are the one who accelerates, the clouds will appear to move relative to you immediately after acceleration. The raindrops will also change their angle immediately. What you are asking ist basically: How can the raindrops change their angle immediately when you accelerate if there is a delay of T for the raindrops before they hit you. The answer is that the raindrops are already in the local space around you and when you accelerate the raindrops emitted before you accelerated will hit you at the new angle. The change in speed for both the clouds and the drops around you is the same, because you are the one who accelerated, therefore you can still use the angle to correctly calculate the clouds currect velocity relative to you. If instead the clouds accelerate and you stand still, only the clouds and all drops emitted after the acceleration will have a different speed. the drops already around you will fall on their old trajectory and you will only see the clouds moving after a delay T when the new drops hit you. So acceleration is not relative and it matters whether you or the cloud accelerates.

If transmitter starts its movement while it's 100 light years away, we will see transmitter movement and Doppler shift effect 100 years later. However, when antenna moves, there is no such delay which is very strange.

As you can see, this happens with raindrops too, so it isn't strange at all.