Although many oil companies own and operate some percentage of their own oil tankers, oftentimes companies elect to lease (or "charter") an oil tanker to transport its oil. These chartered tankers, owned by independent tanker transportation companies, charge rates commensurate with the current tanker market's supply and demand for tanker services.
If a sustained military conflict in the Strait of Hormuz resulted in a significant number of damaged oil tankers, the dynamics of the tanker industry as a whole would undoubtedly be impacted. Simply put, less supply of tankers, due to them being damaged in the conflict, will begin to drive tanker charter rates up. Generally speaking, market rates for tanker services will continue to increase or stay at these heightened levels until either 1) the demand for the tankers decreases or, 2) the supply of tankers increase - a more likely scenario in today's world. The supply of tankers can increase by restoring damaged tankers to operating condition or by delivering new tankers (i.e., building new ones) into the market.
With this increase in tanker rates and corresponding increase in the incurred cost of a barrel of oil for major oil companies, one might expect that oil prices might spike due to the dynamics of the tanker services market. This is not very likely, however, to cause oil prices to spike. This is due to the fact that transportation costs make up a very small percentage of the overall price of a barrel of oil.