Can a tomato pierce a hole in a steel plate if only the tomato is travelling fast enough?

The notion of soft or hard object depends on the velocity of interaction. Water can be soft or hard as rock depending on how fast you fall in (or surf upon).

For a shock, the main thing that matter is momentum. In space, where relative speeds can be very high, a simple bolt can cause serious damage to the ISS, and simple flakes of paint cause deep scratches. So, yes, the tomato would create a hole (and evaporate in the shock).

At high speeds the structure of the material becomes far less important than it is at low speeds. At high enough speeds, the issue is not whether the tomato can retain structure during the impact (it wont), but rather the issue becomes one of sheer mass.

The issue is easiest to see in the tomato's reference frame, where one treats the tomato as holding still, and the plate is moving towards it at a high velocity. At impact, we have a momentum problem to solve. If the plate "wants" to retain structural integrity it needs to deal with the fact that there is mass in front of it which needs to be dealt with. It needs to either:

• Let the mass pass through it, which is very hard given the subatomic forces involved with two masses passing through each other. The result would be catastrophic for the structural integrity of the plate. (I don't think this can happen at realistic speeds, but at relativistic speeds, it might do the job)
• Accelerate the mass forward to the velocity of the plate. This requires a huge amount of force applied to the tomato. The rest of the plate (the part not hitting the tomato) "wants" to keep going at the same velocity. Thus, the outside parts of the plate will not slow down until the information that the impact occurred reaches them (at the speed of sound). The plate has to accelerate the mass of the tomato fast enough to get it up to speed before the difference in velocity of the far edges of the plate and the impacted region is enough to tear the metal apart. Mass is mass. It doesn't matter if its squishy mass like a tomato, or hard mass like rock, if you need to accelerate it, you need a force. That force is proportional to the acceleration needed. Naturally, the faster the plate is going, the higher the acceleration needed will be to keep the whole plate together while the shockwave propagates through the metal.
• Deflect the mass elsewhere, to decrease how much one actually needs to accelerate the mass. If you threw the tomato at a knife edge, it would be much harder for the tomato to break the knife because the tomato's mass doesn't need to move very much to go around the knife. There are still shockwaves, like before, but their effect is mitigated because the shape of the object decreases the determental effects. Of course, at the subatomic level, you'll still see forces that could probably break the knife in half, but we can ignore that line of reasoning because you've chosen a plate. If you think accelerating the tomato to plate speeds in a fraction of a second is a lot of force, just consider how hard it would be to accelerate it sideways even further. In fact, if your plate was perfect (made of unobtanium), it would have to push the tomato sideways at infinite speed to get the tomato mass out of the way. With realistic materials, the plate will react, such as bending, which avoids this absurd infinite case.

In the end, we rely on effects like this for a lot of our ballistic missile defense systems. The whole point of a kinetic kill vehicle is to simply put a mass in the way of the incoming ballistic missile, and let the missile deal with the question of how to deal with all that mass in its way. Ignoring the guidance challenges of an intercept at those velocities, we could fling tomatoes at the incoming missiles and knock them out of the sky.

As Anubhav mentioned in his comment, tomato would break into pieces before it hits the plate. However, to answer the logic of the question such event is possible.

A ping pong ball can rip a huge hole on the ping pong racket if it is fast enough. https://www.youtube.com/watch?v=acRnKnsddwc

Update:

It would make a hole, if you make enough assumptions. As you put it, if we think unimaginably fast (99% of light speed), it would behave basically as light. Since laser cutters can cut through steel plates, I would assume it could.

Yield strength of steel is 215 Mpa, and if you assume a tomato is .2 kg and the impact area is 1 cm^3, then you would need 1075 m/s. (Speed of light is 300.000.000 m/s) These are simple assumptions and basic calculations. I would believe there might be some mistakes.

This simple calculation shows us that the tomato should travel faster than a bullet. (Speed of bullet is approx. 400 m/s)