Ask other staff members how they interpret the mission and objectives. They may have ideas you haven't thought about. Making sure that you understand the reporting structure will help you develop an approval schedule for your project. Inform the director or other persons in the reporting structure that you would like to develop a site for your museum. Find out in advance how much time each approval will require and make allowances in your project plan for those approvals.
Communication is very important.
Your supervisors and co-workers need to know that you are planning to develop this website. It is possible, especially in a large institution, that someone else may have already begun developing a website for the same exhibition or program. Hold a meeting to see if any other staff members would like to contribute to the website development. Ask staff who do not wish to help with the project for content suggestions. If your museum has a website policy, your job will be much easier.