In a word, no.
I have some expertise in the world of Eberron, and I have a degree in electrical engineering, and I am quite confident in this.
We have no reason to suspect transistors themselves would fail in Eberron
I don’t agree that electricity necessarily behaves differently in D&D/Eberron than it does in real life. Magic electricity, maybe, particularly in the form of discrete spells, but that’s not the same thing. A lightning bolt spell is already directing electricity in a direction it wasn’t going to go on its own; that can include into the person inside a Faraday cage, path of least resistance be damned. After all, the electricity in a lightning bolt spell is already ignoring numerous other paths of lesser resistance to get to the target (e.g. the one through the caster), since air is an excellent insulator and the voltage necessary to cause it to breakdown (producing ozone) to allow electricity to flow at any significant distance is very high.
But in the case of a lightning bolt, as opposed to a lightning bolt, I would expect it to work much like real lightning. After all, one of the broadest and most fundamental rules underlying D&D is that things work like real life except when the game says they don’t.
But we also lack reason to suspect that they would know to make one
Even so, that still doesn’t mean Eberron is up to the challenge of producing a transistor: Eberron uses very little electricity. Many of the things we use electricity for, Eberron has magic to handle more directly—lighting, heating, locomotion. A major one for the development of signal theory that directly incentivizes the creation of a transistor or something like it (cf. vacuum tube) does not exist, since telecommunication is handled magically (and Sivis already runs what are effectively telegraph stations all over Khorvaire without any electricity).
Furthermore, the basic purpose of a transistor is to use one electrical signal to control or influence the flow of another signal. It is a control or decision point. A single transistor can be used to make an amplifier; several transistors can be used to create more complex amplifiers, and then things like logic gates, memory, arithmetic operators, and so on. Many millions of them produces a computer. But on Eberron they can bind legitimately-intelligent elementals to things to handle almost all of those properties.
The only reason any kind of electromagnetic theory is likely is because of the existence of the lightning rail, but the conductor stones are the result of the direct application of electrical magic, and the train itself is largely driven by a bound elemental. These do not suggest that Eberron understands electromagnetics with either depth or breadth.
Based on available evidence, Eberron does not seem to have even a 19th-century understanding of electricity (I would be shocked if they had all of Maxwell’s equations, for instance), and the transistor is very much a 20th-century development. I don’t think anyone on Eberron would be able to imagine that these mundane crystals could behave in the fashions that they do.
And transistors are very complex to produce
Transistors are complex crystalline structures that require an incredible degree of control over its purity and the distribution of the impurities that it does have. Even if we believe that magic could achieve those structures (and maybe it can), the artificer would have to know what he was trying to create. Doping ultra-pure silicon crystals with boron, phosphorous, arsenic, or whatever is not exactly an obvious step to take, and particular construction of a transistor is also not one that would necessarily occur to an Eberronian artificer.
In fact, there is no evidence that Eberron has atomic theory, and thus it has no concept of the elements as chemistry understands them. Without knowing what arsenic, boron, phosphorous, or silicon is, isolating and purifying them seems out of the question. And the elements used in semiconductors are chosen for reasons, and without atomic theory you would never be able to deduce them.
How well is electricity understood in Eberron?
Per the above, there must be some understanding, but almost all of it is probably very specific to conductor stones. Not much more than that.
Would anyone understand how to generate it,
Does “get an elemental to do it” count? If so, then yes. If not, then probably not. Considering the existence of elementals, why would they even look for another way to do it? And considering the existence of elementals, there isn’t impetus for an electric engine, which would be the expected progression (figure out how to use electricity to move something → figure out that if you run it backwards, you can get electricity out).
run current through a wire,
Sure, probably. That’s about as basic as you can get. I imagine that Cannith labs have enough metal lying around that the people working on conductor stones and lightning elementals noticed some reactions.
or light a light bulb with it?
If handed a light bulb, sure. Producing a light bulb? Unlikely. It took years for Edison to produce a light bulb that could last half a day, and the tungsten filaments used in incandescent bulbs today were many years after that. Without a pressing need for light bulbs since magic was already amply available and capable of producing light, there would be no impetus for spending that kind of time on the development of a light bulb.
For that matter, an incandescent or fluorescent light bulb relies a great deal on evacuating the bulb/tube, which is not necessarily something Eberron knows how to do. They probably could do it easily enough with the right elemental, but would they? Nothing we know of would give them cause to try it.
But they probably could do an arc lamp, since that’s pretty basic. Arc lamps are extremely loud, extraordinarily dangerous, and exceptionally expensive to run. Almost certainly written off as a novelty.
If so, would there be anyone who would understand things like transistors (particularly logic gates)?
Definitely not. That is a massive step up in terms of development. It’s not something some conductor-stone-researcher is going to stumble upon while playing with the various things lying around the lab. Eberron’s development in numerous fields is much too basic for them to even see a use for something transistor-like.