Does every server in a rack system have an OS installed locally?
Usually, one or more: Today, many physical boxes (a sheet metal box with a motherboard and a few daughter cards, and the motherboard has CPUs and memory) run a hypervisor (such as VMWare or KVM), and run multiple client operating systems in there. A few years ago, the record was about 30,000 guest operating systems on a single machine (but it was an awfully big machine, a multi-million $ monster, and not powered by Intel CPUs).
I guess I'm trying to ask if you can have a computer (a box) without on OS on it that is being controlled by a remote computer.
In nearly all cases, every box (sheetmetal/motherboard/cards/...) will have at least one OS.
In some rare instances (I only know about it in supercomputers), many of the boxes will have an ultra-lightweight OS, which does not have normal facilities (no IO, no login, no user/system split), and are used as pure compute servers. Actually, I don't know for sure that any such machines are still available for sale, but I think some are still running.
There are rare exceptions; in high end supercomputers you can have one "OS image" (single node, since kernel running) that is split over multiple boxes (each with CPU chips, memory, and networking gear). This kind of thing is only available in multi-million-$ supercomputers, but I've seen 4 physical boxes, each with 8 motherboards, each with 4 CPU chips (that works out to 128 chips) that form a single computer.
But again, the very rare exceptions I'm quoting are mostly there to prove the rule: Nearly always, a box (meaning CPU + memory + sheetmetal + ...) will have an OS.
Sorry if this is a dumb question.
There are no dumb question, unless you ask the same question twice (then the second one is dumb).