The organisation wants to construct the building at La Hougue Bie to house machinery and equipment, and to replace a smaller existing store and workshop.
It would be built to the rear of the low-lying old lodge building, in the north-west corner of the site.
The application, submitted to Planning this week, also proposes moving the site's car park away from the base of La Hougue Bie's distinctive mound, towards the entrance. Two trees will be cut down if the project is approved.
Jersey Heritage manages La Hougue Bie for the Société Jersiaise local history society, which owns the site, as part of its remit to care for the Island's historic monuments.
The main feature of La Hougue Bie is the passage grave inside the mound, which was built by Jersey's Neolithic community 6,000 years ago, pre-dating the Pyramids in Egypt by about 1,500 years.
It is regarded as one of the finest passage graves in Europe. Two medieval churches sit on top of the mound.
The work is part of an overall project to redevelop La Hougue Bie which began earlier this year with the construction of a new ticket office.