I would say that yes, it's borderline possible.
The organism should be very resistant to ionizing radiation; a Universe that allows Deinococcus radiodurans to exist would have little trouble in producing this new critter, even limiting ourselves to DNA.
The organism would of course need to consume ordinary matter to replicate; it would therefore find it way more convenient to (also) extract what energy it can from that matter. Actually it's likely that the capability of absorbing cosmic radiation would evolve from the former; the organism acquires matter and this interacts with cosmic radiation, supplying harvestable energy. Once this kind of matter is more and more incorporated into the new generations of the organism, it will "optimize" itself to directly take advantage of the radiation.
An organism could not begin directly using cosmic radiation, since to do that it would need chemicals and structures that would be already pretty complex. It would need to start from simple chemical reactions, and evolve to first defend against, then make use of the radiation. The most obvious evolution pressure line would be if the defense takes the form of some radiation-hardy chemical, which is transformed on impact generating much higher-energy compounds that would be initially disposed of. Afterwards, any mutation allowing to extract energy from such compounds would have an enormous evolutionary value.
This is not unlike what already happened to radiotrophic fungi, which produce a special kind of melanin that is capable of harvesting energy from gamma radiations. It would be more difficult (!) for our hypothetic organism, in that cosmic rays are much more penetrating and energetic than gamma rays and therefore require proportionally more massive interception.
And, of course, the organism would need to live somewhere with a lot of cosmic rays.
One possibility would be some sort of low-temperature outer space slug capable of accreting a "black ice" carapace. It would migrate to the safe center of cometary nuclei to spawn, then come out when the carapace is thick enough to both defend the young slug and supply it with energy. The slug would need to have a very low metabolic rate, and be based on a totally different organic chemistry than Earth organisms.
There's something along these lines in "Camelot 30K" by physicist Robert L. Forward, where a (macroscopic) lifeform is presented that harvests atomic nuclei transmuted by the impact of cosmic rays on distant comets.
"The initial energy source for the kerac civilization comes from the background cosmic radiation, which creates long-lived excited molecules and free radicals in the ice over long periods of time". He drew an arrow leading to another box that he drew in the rotund shape of an iceworm. "Out on the farms, the iceworms and young heullers tunnel through the untouched ice around the periphery of Camalor. They extract free radicals frozen into the pristine ice and use them as a source of energy to grow and continue tunneling. Those long-lived free radicals are the first level energy source of the kerack food chain, since the iceworms and heullers are harvested as meat. While the iceworms are doing that, they also extract from the dirty ice all the unstable radioisotopes that the dirt and ice contain."
(While different organisms are named - iceworms, heullers, keracks - it is posited in the book that they are actually all alternate phenotypes of a single organism, the kerack hive drone, not unlike what happens with bees).
