As of june 1, The 1940 census was searchable by name for about 20 states, and the District of Columbia.
Here is what is available for each partner site:
--Ancestry: Delaware, Maine, Nevada and Washington, DC. A chart on the 1940 census page lets you see indexing progress.
--FamilySearch: You can search 14 states/territories by your ancestor's name: Alaska, Arizona, Colorado, Delaware, Florida, Idaho, Nevada, New Hampshire, Oregon, Vermont, Virginia, Kansas, Utah and Wyoming.
--Archives.com: same states as FamilySearch. To access the unindexed portion of the census, this site sends you to the National Archives' 1940 census site.
--FindMyPast.com: same states as FamilySearch, along with Hawaii, Louisiana, Mississippi and Montana.. Alaska, a territory in 1940, is missing from the color-coded map on the home page.
--MyHeritage: there is a searchable name index for Rhode Island, and a partial name index for New York.