Not a gas giant, but not a rocky planet, either. Rather, a planet with just light elements, like the first 3 rows of the periodic table, or so. You probably need carbon or something similarly flexible to get life at all, and iron and other trace elements are necessary for earth-based life, but it is at least plausible that exo-life could use something other than iron to carry oxygen. Or, there are only trace amounts of iron on the planet (iron is a rare-earth metal) and most of it is bound up in living things. This prevents you from making anything we would recognize as cars and trains and planes, as their sturdiest construction material would be something like wood (if you have stone, then you have Si, and if you have Si-based electronics, you probably have other heavier metals like Fe and Ni, and thus, cars).
So, how do you get electronics? Most conductors we use today are metals, Cu to Fe to Au. Well, this is where it gets tricky. This planet would basically need carbon nanotube-based electronics. It turns out that you can make carbon nanotubes using very low tech (pretty much just burn any organic matter and you will get some by pure chance). But how you turn that into electronics without the benefit of metals requires a bit of creativity. On the other hand, without metals to begin with, an alien would be forced to think differently anyway. You could also use conductive polymers, or go the biological route and use doped water to make wires. Consider that we already have the most energy-efficient computers in the known solar system sitting inside our heads.
So, carbon nanotube electronics would be the cutting edge of technology on this planet, while neuro-electronics would be the default mode of computation. They would most likely have bio-computers rather than what we think of as hardware. This would also help explain the energy sources. They would lack the Si and rare earth metals necessary to make PV solar power, as well as the radionuclides for nuclear (no U, Pu, Th, etc.). They would have fossil fuels, but nothing sturdy enough to build boilers, let alone internal combustion engines or gas turbines. The best internal combustion engines you can build with light elements use the ATP cycle and power your muscles.
The question then becomes not: "Why do they use animals?" but rather: "How did they get electronics?" At some point, they could likely develop counting and mathematics on their own, and could build basic mechanical computers like abaci from just wood. They may even build primitive machines like a TinkerToy computer to play tic-tac-toe using wood and animal parts (bones, etc.). But at some point, they would realize the need to build fast switches in order to make a truly general computer. This is where things get ugly...
In the course of hunting, they will notice that sometimes animals move after they are dead, and they will wonder what causes this movement. Eventually, they will realize that nerves are conducting signals to muscles, and that nerves are actually wetware switches. However, they will also realize that nerves degrade very quickly once their host creature has died. So to exploit the natural wetware electronics, they will need live creatures.
In the beginning, they will select animals which are a convenient size for handling, like small rodents. They will figure out how to use stimuli on one end to trigger behavior on another. For instance, they might notice that pricking the paw of a rat causes its tail to move. Or maybe if you price one paw lightly, the tail doesn't move, but if you prick both paws, it does. And now you have the crudest bio-switch (a kind of logic gate), whose energy source is almost any organic matter, but runs reliably on vegetables. You just need to chain them together by connecting the tail to needles on the paws of downstream rats, and you can build a complete ratputer to do simple calculations.
While this sounds terribly cruel, keep in mind that you can't build a decent-size circuit with rats, and these scientists/engineers will figure that out pretty quickly. Also, such a ratputer would be very power hungry (in the most literal way) and produce a serious waste problem. So you need to miniaturize. You then move to smaller and smaller animals, until you can work with, say, ants. Because insects already communicate with each other, you may actually be able to train them to use multiple modes of transmitting information in a computational network. You could use mechanical stimulation for precise calculations (a leg moving could tickle the antenna of a downstream ant), and pheromones for high-level signals. Also, the ants could be explicitly trained to produce certain behaviors (being fed one sugar drop makes its 4 leg wiggle, and being fed a different flavor drop makes its 5 leg wiggle).
Over many generations, they may actually be able to create a breed of insect which is specialized for computation! The legs may naturally lock together, forming quasi-mechanical circuits, and the ants may become specialized for their task by increasing signal propagation rate, accuracy, and even adding new functions which would not occur in the wild. Eventually, the legs could fuse together, creating a kind of hybrid creature in which each ant brain is like a local distributed CPU, operating on a mesh of power droplets raining down from above, and releasing waste to a collector plate below. Multiple "compute boards" could be stacked into a 3D space to provide more powerful computation. And since the antennae and eyes and carapace would become redundant, these features would be bred away until the ants eventually formed nearly pure neuron networks with the minimal internal organs to support them, complete with interlocking feeding/respiration/waste plumbing so that they can be packed densely.
The limiting factor for all civilizations we know of is energy. Technology only moves forward when energy availability increases. Fire to cook food releases more nutrients for humans. Food for humans powers population levels. Fire to melt metals provides tools for building, hunting, and warfare. Fire contained in metal can be harnessed to power machines. And nuclear fires contained or uncontained can be harnessed to do planet-altering things.
If you prevent energy from being harnessed, then you can't plausibly build vehicles or computers at all. So forbidding fossil fuels is both implausible and unnecessary. The only way to prevent earth-like technology is to take away the key ingredients which underlie our tools: metals like Si, Fe, Cu. These give us concrete, steel, and wires. Whatever is left can be used by biology to build alternatives.