Who works on strategy
The cheapest answer: everyone in an organization. From C-level to middle management to workers. Though levels of decision-making might vary, each decision should be made with awareness towards the strategic initiatives of an organization.
The more differentiated answer:
You will probably find a core strategic team that is responsible for deciding which options to work on, where to respond to changes in the environment.
They alone will not be sufficiently equipped to bring strategic options to live. They need an extended team of people stewarding options and actually working on them.