Are living organisms deterministic?
In the sense that most people mean, I'd say no because quantum mechanics isn't deterministic (at least the way most people think of "determinism").
Ultimately, to be more specific, I think this depends on your interpretation of quantum mechanics, and how reductionist you are.
The best you can do with reconciling quantum mechanics with determinism is taking a particular (admittedly relatively popular) interpretation of quantum mechanics (the many-worlds interpretation) which is "deterministic" but not in the sense you were probably thinking of, or in a sense which is very useful.
Reductionism, which is generally the working mode of thinking for scientists, implies that if quantum mechanics & physics is deterministic, then chemistry and biology are too because they are built only out of pieces of nature that are fully described by quantum mechanics and nothing else. If you want to be controversial, you can suppose that there exists "strongly emergent" phenomena which fundamentally can't be traced to quantum mechanics & physics (e.g. a soul which is not described by physics). This generally isn't very popular in experimental sciences because 1. it's not very useful and 2. these ideas generally don't have a great track record.
More on the quantum mechanics bit:
Quantum mechanics gives describes a system of particles and their evolution in time in terms of probabilities, not "deterministic" events, as classical mechanics would. And there's good reason (see Bell's inequalities and Bell tests) to believe that this probabilistic description is as fundamental as you can get for these things. The purest description of the universe is in terms of probabilities.
Probabilities seem manifestly non-deterministic. However, if you subscribe to the many-worlds theory then this is still in a sense deterministic, as quantum mechanics (or a similar theory) would correctly (and deterministically) describe the evolution of the state of the universe. Locally (i.e. in our part of the state which can't really observe other parts of the state) things look probabilistic, but if you could somehow observe the state of the whole universe, you'd deterministically predict how the whole thing evolves using quantum mechanics. If you don't know what I mean by this, I'd start by looking up the concepts of superposition and entanglement in quantum mechanics. As of yet I don't think there is anything testable about the differences between this and other interpretations of quantum mechanics (e.g. Copenhagen), so many people don't like reading into these things too much.
Note that systems are not deterministic, dynamical theories are. So you can look at a dynamical theory, and see if "deterministic" is an attribute of that theory or not, but all you can do with a system is ask "do I have a theory that predicts to satisfactory accuracy what that system does?" Here's the key point: if the answer to that is "yes", it still does not make that system deterministic, it merely makes the treatment that you are successfully using deterministic. At a higher level of precision, it is invariably true that your theory won't succeed any more, so what are you to say then? In the case of living systems, no deterministic theories have ever been successful in treating their behavior to any degree of accuracy. So the above gives us two independent reasons for rejecting the claim that living systems are deterministic: 1) we have no test if a system is deterministic, only if a theory is, and 2) no deterministic theories have ever worked well in all situations for any living systems. I suppose this allows anyone to believe what they like, but their evidence will be poor.