Can I legally run a salon from a garden room?

In most cases, yes, and plenty of people already run a salon business from home this way. The question is scale rather than principle. Permitted development covers a garden building used incidentally to the house, and a single operator seeing a handful of clients a day generally sits inside that, so no planning permission is needed. Where it changes is when the property starts behaving like commercial premises, with cars arriving all day, staff who do not live there, signage at the gate and appointments running late into the evening. At that point, a council may treat running a business from home as a material change of use.