This is somewhat different to this question on the possibility of biological fusion reactors as I am curious if a living FISSION reactor is more plausible.
The short answer is NO - a biological fission reactor is also a flawed design and it's going to cook your biological organism so it's just not going to happen. The energy cost alone to reach break even point is far beyond the biological framework we live in to store and release in such concentrated amounts.
That said, it is perhaps important to note that biological organisms are already molecularly driven even if that doesn't provide the same power as we would get if we were atomically driven.
Terrestrial biology (at least) stores and releases energy at the molecular level. That is to say, plants take molecular compounds like CO2 and H2O and via an endothermic reaction called photosynthesis, change the molecules to different ones like O2 and carbohydrates, which take more energy to form, hence storing the energy taken from sunlight in a molecular form. Plants use some of this for their own metabolism, and animals like us then eat the plants, taking their stored carbohydrates and oxygen from the atmosphere and generate an exothermic reaction that converts them back to CO2 and water, releasing that energy inside us for our own needs.
The energy stored and released during these processes is pocket change compared to fission let alone fusion, but as it turns out we don't really need all that much energy to survive; even the human nervous system seems to operate at around 0.07 volts, meaning that you could run around 21 people in parallel on a standard AA battery if you had to (but for how long I'd need to do more research to say).
Bottom line is that we don't need atomic power to drive our biology and the energy storage capacity of our current biological design would render us incompatible with it in any event. It turns out that being molecularly powered rather than atomically powered gives us all the energy we need and is much safer and sustainable for us into the bargain.
Well, first let's see what the creature has to have.
A sizable chunk of Uranium. After all, you can't have fission without fissionable materials. Unfortunately for your animal, Uranium is not very common, meaning that your biological reactors would be restricted to places like Utah's Factory Butte, where there are (reasonably) large amounts of Uranium on the surface.
Some way to safely process Uranium. While it is possible to process Uranium without a complex lab, it is extremely likely to cause irradiation of tissues, which is no bueno for your creature. The only reasonable way around this is to have an "access port," in which people can dump preprocessed Uranium fuel rods.
Radiation shielding (since you don't want the radiation killing your creature.) For this, you pretty much only have one option: your creature somehow processes lead in bulk, and then somehow cause lead deposits to build up around the "reactor core." This is almost impossible to have evolve, so you pretty much have to make your creature a GMO.
Coolant systems. Not only will your reactor be producing a lot of radiation, it will also be producing huge amounts of heat. After all, that's what a reactor is supposed to do. In this case, pretty much the only option is to have your creature immerse itself in a fast-running river and then somehow ditch the heat safely. Although not as hard to evolve as radiation shielding, it would probably be another GMO feature.
Using the energy. While it is possible to make the reaction core part of your creature, there is simply no way for your creature to harness the energy in any meaningful way. While it is possible that it could use thermoelectric power generation, this is another complex process which cannot come about naturally. In addition, it would be very hard for your creature to make this energy usable, but that can be handwaved.
All that being said, while your creature is impractical in real life, it is a very cool idea as a minor aside, and the problems can be handwaved. That being said, I very much would not give these creatures a major role in your story, as then your handwavium would break down.
Can an organism evolve to use a concentrated mass of radioactive material to superheat a pressurised fluid and use that fluid to turn an organic turbine which rotates a permanent magnet inside a coil of conducting metal to generate electricity? Almost certainly not. Does that mean an organism can't make use of nuclear fission to energise itself? Not at all.
On an atomic scale, fissile elements are basically tiny bombs: they're happily sitting there as one element in whatever molecule, then bam, one or more high-energy fragments ricochet off in a random direction potentially screwing up another nearby molecule, and the original atom is now two totally different elements with different chemistry. We make fission reactions more energetic by concentrating fissile material so that the fragments are often captured by other fissile atoms which prompt them to react in turn, creating a chain reaction; but even in its simplest form a pile of fissile material releases energy by capturing the high-energy fragments in something and thereby getting slightly hot, no feedback loop required. The challenges for an organism safely utilising this energy source, then, are:
- Extract fissile materials from its food and concentrate them
- Deal with the heavy element byproducts of the fission reaction
- Deal with the highly-energetic (usually neutron) fragments and convert their kinetic energy into heat, then convert that heat into a more useful form of energy for the organism
Biological organisms have evolved to manipulate chemistry on a molecular level; they can catalyse pretty much any viable chemical reaction, concentrate particular atoms on one side or other of a boundary, and so forth. It if were evolutionarily favourable for the creature to extract, say, uranium or plutonium from its food, it could do so with no more difficulty than mammals extract magnesium, calcium or potassium ions on Earth. Filtering atoms by isotope is much more challenging, but it's not impossible that an exciting protein could evolve that would bind fractionally more strongly to one isotope than another. Problem 1 solved.
Problem 2 is equally plausible: while the exact byproducts of a fission reaction are random, they'll cover a small range of elements, and if the organism is 'expecting' them then there can be mechanisms for controlling and excreting them if they're not usable. Poisons are only poisonous if the organism isn't adapted to deal with its presence: arsenic is highly toxic to humans, but plenty of terrestrial organisms are evolved to tolerate high concentrations of it.
The main challenge an organism would face is actually doing something useful on a molecular level with the energy released by the fission reaction. By dissolving the fission products in water or a more powerful neutron capture liquid. Boron is an excellent neutron absorber, so boric acid might be a good choice here. The net effect of the nuclear reaction, then, is to turn cold boric acid into hot boric acid. Then there are already terrestrial organisms which survive on reactions which exploit temperature gradients like those found at hydrothermal vents: the organism could either exploit those reactions themselves, or live in symbiosis with a bacteria that did.
In short, while you're unlikely to see an organism evolve to set up the highly unstable, low-entropy, complicated arrangement of materials that humans would consider to be "a fission reactor", fission is an energy-releasing process, and an organism evolved to harness that process in a more relaxed, stream-like, organic way, is perfectly possible.