Basically, for devices identifiable as electronics to be possible in a similar manner as the technology's progression on Earth, you need the following to be common on the planet:
- A ferromagnetic metal (preferably iron, nickel or cobalt if you must)
- At least one Group 11 transition metal (copper, silver, gold)
- At least one post-transition metal (tin, zinc, indium, aluminum)
- At least one noble transition metal with a high melting temperature (tungsten, rhenium, osmium)
- At least one metalloid (silicon, germanium, boron)
- A gas that is chemically inert even at relatively high temperatures (noble gases mainly; neon, argon etc)
If you have all of these, you can at least get to the vacuum-tube era of electronics. Progressing beyond this level requires a wider variety of "rare-earth metals" and other substances, but you can get to a basic electronic computer if you have a relatively plentiful supply of at least one of all the above options. If you have a basic idea of why they're needed, skip the next section, otherwise keep reading.
Ferromagnetism is a requisite for the efficient generation of electricity. Moving a magnet within a coil of wire is how about 97% of the world's grid electricity is originated (photovoltaic solar power, the most common non-inductive generation method, produces only about 2.5% of infrastructural power, even taking replacement of actual grid power with roof-mounted systems into account). Iron is the ideal candidate for a ferromagnetic metal useful to humans, and is pretty easy to find on Earth. Nickel and cobalt metals and their alloys are magnetic too, but these happen to be ridiculously toxic to human biochemistry, so sleeper-ship colonists reduced to "off-the-grid" tech wouldn't want to be anywhere near an open-air smelter of any ore containing either metal. Neodymium is not itself magnetic, but in a crystalline alloy with iron it boosts iron's potential to be magnetized, thus allowing stronger magnets with less material. It would be nice to have but isn't critical.
With the basic material to induce electric current, a soft, ductile, highly-conductive but nonmagnetic metal is what we tend to induce that current in. Copper is one of the first metals to be harnessed by humanity, and fairly plentiful on Earth (though our demand for it in industrial and post-industrial economies is making it a more expensive commodity of late). Metals in the same elemental group are rarer on Earth, to the point of being stores of value, but on a planet where gold was very common, that would do many of copper's jobs better than copper itself (copper would be relatively lighter and a little more durable, offset by gold's natural corrosion resistance reducing required maintenance). Silver's closer to copper in most of its properties (including unfortunately that it oxidizes or tarnishes easily) and again if relatively common on some new planet it could do most of copper's jobs, but silver's commonly found
Having generated and conducted electricity, you need to use it. Resistive emission ("intentionally wasting" electrical power to generate heat and light through electrical resistance) was one of the first discovered uses of the technology, closely followed by inductive motion (using current to generate a magnetic field that pushes or pulls another magnet); heating and lighting filaments and inductive coil transducers (motors, solenoids, speakers/microphones) remain the two largest classes of electrical "doers" on the planet (even as a much more efficient alternate form of light emission using semiconductors has become the most common form of light generation).
While Group 11 metals work well for induction as they conduct electricity very well, that same conductivity (and their low melting points) means that they're relatively poor at resistive applications. Iron, especially as a high-carbon steel alloy, is a passable resistor especially for heat generation, but it's relatively inefficient at generating light (the "color temperature scale" of room light is calibrated based on the spectral emissions of iron heated to the prescribed temperature, giving you some idea how hot iron has to be to glow white and how much power that takes), and the heat involved speeds corrosive oxidation. Various organic compounds like graphite make good low-power resistors, but turn up the voltage too much and they'll literally explode. The generation of electric light involves forcing electrical current through something that does not conduct well, but is also not physically or chemically altered by the electricity flowing through it or the heat it's producing. The best such element we have in what we'd call a plentiful supply is tungsten, a member of the "refractory group" that also includes metals commonly alloyed with iron to produce stainless steel, including vanadium, chromium and molybdenum. The platinum group is another good area of the Periodic Table to have available, with rhenium having the second-highest melting temperature after tungsten, and being very corrosion-resistant. Platinum itself is used in PCBs, though it's not a critical requirement.
A low-melting-point post-transition metal has a number of uses in electronics, for instance as a solder material to form high-continuity wire junctions. These post-transition metals are also valuable as fuses, as resistive heat eventually exceeds the melting point and breaks the connection. Tin's a very common choice nowadays, replacing lead especially in plumbing for what should be obvious reasons. Aluminum has a number of uses for conductivity of heat and electricity in electronic devices, however at its actual melting point it will also burn readily, and its dissociation temperature (where it will separate from the oxygen and become a useful metal) is ridiculously high, making aluminum a precious metal until the Bayer and Hall-Heroult processes made aluminum refinement commercially viable by WWII. More exotic ones include indium, which you can melt in a glass beaker over a Bunsen burner (or even a good hotplate) and is very rare on Earth.
The metalloids have several uses. Primary among them is that most are dielectrics; they do not conduct electricity (at least not below an arbitrarily high voltage not typically relevant to benchtop electronics), but they can become polarized by electric charge, and by so doing allow electrical interaction to occur through them. This property, especially of silicon (one of the most plentiful elements on Earth) has led to a wealth of uses in semiconductors, from capacitors to transistors. The development of germanium- and then silicon-based solid-state transistors is what enabled the proliferation of modern electronic devices. However, and earlier and more primitive, but no less critical, use for metalloids is as glass. Glass containers are airtight, nonconducting and non-magnetic materials with relatively high melting temperatures, making them very useful for containing high-temperature air-sensitive components like charged plates and filaments. Modern electronics would not have been enabled by the MOSFET solid-state transistor if it had not been preceded a generation earlier by the vacuum-tube triode transistor, and that in turn required the relatively simple but deceptively difficult artisan skill of glass forming to have been developed and refined beginning before the Renaissance and proceeding into the 20th Century.
I've hinted at uses for inert gases, but to be clear, if you don't have an inert gas, a lot of things become a lot harder to do, because your "vacuum tubes" have to have a true vacuum, and this shortens the life of the electronic elements as the metal literally evaporates into the vacuum under heat and electrical excitement, and deposits onto the inner walls of the glass (a process called sublimation, which we have since harnessed to produce electronic components by etching circuits into layers of sublimated "thin film"). Argon is the main such gas we use, krypton would work just as well but is relatively rarer on Earth and is prized for use in light-generating equipment for its multi-spectral pattern under electrical excitement. Nitrogen, the most abundant gas in Earth's atmosphere, works well enough at "normal" temperatures to exclude oxygen from a volume of space as a blanket, but when you get into the high hundreds of degrees it starts becoming reactive (the Haber process for synthesis of ammonia starts becoming favorable around 400*C, though high pressures and a metal catalyst are also required).
So, if you have a sufficient selection of all of these materials, you can fashion rudimentary electronic components up to and including early 50s-era computers. To progress further, you need the MOSFET, which requires gallium or boron for P-doped transistors and phosphorous or arsenic for N-doped transistors, in addition to either silicon or germanium as the body material. That gets you to printed circuits, allowing progressively smaller amounts of miniaturization. Other elements, reasonably common on earth but not often in the combinations we use them, are used in modern electronic processes, including chlorine trifluoride which is used as a cleaning reagent for thin film equipment (at which it undoubtedly excels; it's the most vigorous oxidizer known to modern chemistry, so nasty to handle even the Nazis had second thoughts about weaponizing the stuff).
A planetary environment lacking significantly in either iron or copper would dramatically inhibit the natural development of electricity-based technologies similar to our own. Of the two, iron deficiency would also inhibit multi-systematic life forms as we know them; a dearth of iron would preclude the popular development of iron complexes like hemoglobin for oxygen capture and transport, which on Earth would have stunted eukaryotic life at the plant level. Sentient plants have been sci-fi fodder for a while, but at least on Earth the energy reserves and transportation mechanisms of even the very largest or most energy-dense plants are far too low and too slow to fuel the more complex neural networks found in animal life.
So, making copper an impractically rare trace element would be the most likely way that life forms we'd recognize as such could develop to sentience, but which would limit the development of modern technology as we know it. In fact, without an easy-to-smelt and economically-useful metal, such a civilization would likely be trapped in the Neolithic era of human development.
Which, as we know from our own history, is a totally valid steady-state; the Mayans, Incans, Aztecs, Pacific Islanders and other New World cultures managed to develop quite advanced civilizations and technology based around stone and organic materials, with little if any knowledge of metallurgy. However, again evidenced by the fate of New World native civilizations, carving stone only gets you so far, and New World knowledge of forming and using metal was relatively limited.
The most common metal known to be commonly worked by New World civilizations, rather ironically, was gold, whose ore is basically the raw metal with impurities that are fairly trivial to separate, and is relatively easy to melt and to work. These same properties, however, make it less useful as an infrastructure metal even if plentiful; it's not a very strong or stiff metal, and it's dense (#8 on the periodic table overall, out of all elements we've been able to accumulate a cubic centimeter of at one time and place for an empiric test). Copper would have been known, but its relatively higher rate of corrosion (not to mention the higher difficulty and inherent hazards of smelting copper out of the higher sulfur- and arsenic-containing ores) would have made it less economically viable than in Europe. The relative dearth of zinc (most plentiful source in the Americas was thousands of miles northeast of the Aztecs' widest range, in the Mississippi and Tennessee River valleys) and of tin (currently mined in Peru, but with modern methods of extraction and refinement not developed until the Industrial Age) would have limited the New World's knowledge of alloying, giving Europeans about 4,000 years' head start including about 2500 years' experience with iron and eventually with steel as of when the conquistadors first set foot on continental South America.
This lack of knowledge and of available materials also meant that New World civs, much like any hypothetical extraterrestrial race developing on a similarly-deprived planet, would have less opportunity to discover electrochemistry and start connecting the dots. The generation of static charges is the most commonly-identified early source of harnessed electricity, however it was Alessandro Volta's invention of the "voltaic pile", a copper-zinc metal-acid cell, that produced enough sustained electric current to enable focused scientific study. This would lead to Oersted's accidental but repeatable discovery of electromagnetic fields, in turn leading to Farraday's laws of electromagnetic induction which fully link electricity and magnetism as a unified theory of a fundamental physical force, not to mention powering the world as we know it.