"回る" is an intransitive verb, and therefore not appropriate the way you are trying to phrase it.
In Japanese, sometimes there are two types of the same verb often referred to as transitive and intransitive verbs. The difference between the two is that one verb is an action done by an active agent while the other is something that occurs without a direct agent.
So you can say, "椅子が回る" to describe the seat turning, but not turning it. If you had said "椅子を回る", it would rather mean "go around the chair", but that is a different story.
"回す" is the transitive counterpart to "回る". "回す" takes the particle "を". You can phrase it in the following way:
"させる" is a causative verb, which can essentially mean "make" or "let" someone/something do something.
Verbs conjugated into the causative form are used to indicate an action that someone makes happen.
"回転する" would be what the seat is doing, while "回転させる" would be what you are doing to the seat, or making it do.
Lastly, I would prefer to phrase the request as following:
The why is mostly preference, but "ください", albeit humble, is imperative. Using potential form is also a pretty standard and soft way of requesting something. You can make it even less direct by using phrasings suggested by nodakai and Shoko.