You could model this a few ways. You could say a creature exports its entropy by eating food, or you could say, more accurately, that the entropy was created when the energy was stored in chemical form in whatever substances the creature eats.
If there was some ultimate limit on the ability of creatures to communicate information forward then the aging process would apply to the entire species, to life itself, rather than to individual organisms.
You can think of yourself as an organism that was created a few years ago by your mother and father, but as a chemical process, you are 4 billion years old, give or take a few hundred million years. There has not been a single second, in the last 4 billion years, when the chemical reaction that is you was not working to propagate itself forward. If entropy applied to you, then surely you would not have lasted 4 billion years?
I don't believe it's necessarily due to entropy but the process by which cellular division takes place involves a copying of the host cell's DNA and that copying process is not 100% perfect. It is remarkalably accurate, but given enough time, even with all the systems in place to mitigate and repair errors, they will occur. Most will simply cause that mutated cell to die but it's similar to the 100 monkeys on 100 typewriters at scale. Eventually an error will occur that makes a cell immune from apoptosis.