One of the simpler endothermic reactions is the dissolution of various salts in water (or other solvents). Ammonium nitrate demonstrates this, as found in commercial chemical cold packs; these work (or once did, the manufacturers may have changed chemicals over time) by packaging dry ammonium nitrate prills with a capsule of water, and are activated by rupturing the water capsule and kneading the pack to spread it around. When the ammonium nitrate dissolves, the increase in entropy is sufficient to drive a reaction that lowers the temperature of the solution quite markedly from the ambient temperature of the water and dry salt.
There are a number of other simple inorganic salts that exhibit this to a lesser extent (I suspect all do, but I'm not a physical chemist and haven't checked).
Nitrates are within the possibility of living organisms to create; my partner has "denitrate rock" in an aquarium, which is misnamed because it harbors bacteria that, in turn, convert ammonium ion in solution to nitrite, then nitrite to nitrate, and finally break down the nitrate to nitrogen and oxygen. The first two steps are all that's needed here to give an organism internal production of nitrate from ammonium (produced by other bacteria from urea), followed by a process similar to the ion pumps in the kidneys or reverse osmosis in the intestinal walls to dehydrate the salt for storage (producing heat slowly that can be gotten rid of the usual way, by conduction or evaporation).
Once the dry ammonium nitrate (or other salt) is stored, when the animal needs a quick burst of cooling it need merely introduce water (such as lymph or intercellular fluid; it need not be particularly pure due to the very high solubility of ammonium nitrate) to enjoy a fairly sudden burst of cooling without having to wait for the environment to take up the heat.