As a foreigner, can I legally own property in Phuket?

It depends on what kind of property you are looking for. You can legally own a condominium as a foreigner (Condominium Act) but it is a little bit more complicated with houses and villas.
As a foreigner you can not own land in Thailand but you can own the house that is standing on that land, so you would have to lease the land when buying a house.