What is the small room most businesses have at their entrance called?

The difference between a foyer, lobby, vestibule, and reception depends upon function, size and structure.

A lobby and foyer are typically synonymous, but lobby implies that it is a larger size. For example, hotels and theaters have lobbies, but houses do not. A lobby does not need to be at the entryway (although it typically is). Lobby derives from old Latin lobbium, meaning gallery.

A hotel or theatre could have a foyer, but that would typically be a small room off of the entryway. A house can also have a foyer. Foyer derives from focarium, Latin for center of focus.

A vestibule is also small room at the entryway into a building. It is derived from the Latin vestibulum meaning forecourt. It would be rare in AmE to describe a house as having a vestibule. It is a dictionary synonym to the other two, but in common usage I would say it is related more than synonymous.

Of the three there may be dialectal differences governing which to choose. For me, in (New York) AmE, a lobby is larger than a foyer,which is larger than a vestibule. But, I gather they can be used fairly interchangeably in other dialects.

Reception (short for reception desk or center) is the last of these, and it is not an exact synonym. Reception is typically located in one of the the three above. It is the place where people who enter are greeted, typically by a receptionist.

I agree with DavidM that the most technical word for it is vestibule, however, most people where I live (central Canada) would call that little room in an office building, containing the inner and outer doors, an entranceway.