A argument against this is that companies basically engage in a survival of the fittest contest

Counterpoint from the archives: "No Evolutions for Corporations or Nanodevices"

Are markets therefore a special case of evolution constrained to a certain class of agents? Or do the results they produce warrant them being included in a new family of optimization processes besides evolution and intelligence[?]

I would say No and Yes, respectively: it is not useful to think of markets as a special case of evolution. It's true that an English-language description of both processes is likely to involve the word competition, but the underlying details are very different. Evolutionary theory concerns itself with things that replicate and the consequences thereof, whereas economics concerns itself with agents that trade and the consequences of that.

Perhaps the condition is that the process must be able to work without the "use" of another process.

I don't agree with this condition. Microeconomics does require agentlike components: market participants are assumed to have preferences that they try to satisfy subject to some sort of budget constraint; if you don't have something that at least approximates that structure, then you don't really have anything we would want to call a market.

And yet despite being made out of smaller optimizers, it does seem fair to say that markets are a kind of optimization process in the sense that the system as a whole produces highly nonrandom outcomes. We have theorems that say if you assume that utility-maximizing agents with complete, transitive, continuous, monotonic, convex utility functions over some finite set of commodities trade under perfect information, then they reach a Pareto-optimal result where price is proportional to marginal utility, &c. Of course the assumptions made by such models are ludicrously unrealistic when it comes to actual markets made out of humans, but they illustrate the point that market forces are doing something interesting that isn't a simple property of any of the market's component agents; you could say it is emergent (in a nonmysterious sense).