Can you have vice president in a company if there is no president?

Yes, that is possible. Companies have either an owner or a board of directors who appoints positions such as president and vice president; unlike the US federal government, there is no automatic promotion of a vice president to president in the event that the office of the president becomes vacant for whatever reason. When a company has no president, the owner or board of directors has to appoint one, and they do not necessarily have to promote the vice president. And there could be a period of time when the new president has not yet been chosen and there is still a vice president. It would be an unusual situation but certainly not an impossible situation.