The answer comes down to skills and operational requirements. If your online strategy is important to your business then this might reasonably be regarded as a core competency.
Any business that has a significant online strategy needs (at least) to maintain someone in-house who is capable of designing that strategy.
If that strategy includes an online application portfolio of non-trivial complexity then they need someone in-house who is capable of designing or at least specifying that application portfolio.
If the application needs to evolve responsively (i.e. in 'Internet Time') then the business needs a process that allows these changes to be specified, implemented, tested and deployed quickly.
For static content, a good content management system might be sufficient for this. If you have an application portfolio you may find that outsourcing the development significantly impedes your ability to manage quick evolution. This implies that outsourcing the development process is likely to compromise your ability to fulfil point (3).
The chances are that you will only be able to find the skills to fulfil the requirements in point (2) in someone with significant current internet application development and architecture experience. This implies the presence of a development team that is (at the very least) capable of developing prototypes.
Finally, maintaining an e-business strategy without a technical backing is a very risky proposition. An online strategy designed by someone without technical skill is at risk of failing due to unanticipated technical considerations or even fundamental design flaws. Thus, in order to fulfil the requirements of point (1) the online strategy design process must have input from someone who is technically literate. This could be outsourced to a consultancy if points (2) and (3) are not key operational requirements of your online strategy. Otherwise this technical expertise should be available in-house, which implies the existence of a development team with at least the capabilities to fulfil the requirements implied by point (2).
This development team could consist of a mixture of in-house people, contractors or people working on secondment from a consultancy. The make-up of this could vary based on the consistency of the work, requirements for specialised skills (e.g. predictive analytics) and work volumes.
Rick Chapman's 'In Search of Stupidity' chronicles software marketing and management disasters made by non-technical management. His thesis is that a technology company run by non-technical management is at a serious risk of management making a catastrophically poor decision through lack of understanding of the technology. It might be argued that this principle extends to any significant technology related aspect of business strategy including strategic online presence.