The short answer is yes. There is no law in the United Kingdom stopping you from owning body armour.
Body armour is used privately by individuals who undertake work where there is a perceived risk, such as being a security guard or alarm responder.
Usually these individuals will wear covert body armour that is designed to be worn under clothing to avoid attraction attention, however overt armour has become incresingly popular especially within local authorities.
Today many council employees such as taxi rank marshalls or litter enforcement officers will wear highly visible vests so the general public can see them from a distance.
It is worth noting that body armour is illegal to wear in some countries so if you intend to travel with your armour check the relevant laws first. For example, in Australia it is illegal to possess body armour without authorisation in certain territories (South Australia, Victoria, Northern Territory, ACT, Queensland and New South Wales).
In certain Canadian provinces (Alberta, British Columbia, Manitoba and Nova Scotia) a license is required to possess body armour, though there are no such restrictions in the rest of the country.