When did the government switch from manuscript to typeset documents?

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It does not appear there's any definitive answer to this.  Assuming you mean typewritten documents (as opposed to mass-printed documents, as the printing press long predates the U.S. Government), this varies widely by agency, office, location, resources, and individual circumstance. Government use of typed documents begins in the late 1800s, following the introduction of commercially available typewriters.  Multiple National Archives staff members contributed their observations:

One NARA staff member notes:

Sometime in the 1890s Congress switched from handwritten to typeset for public acts:

Here's the handwritten Sherman Antitrust Act (1890)

and the typeset Declaration of War with Spain (1898)

Another has observed wide differences between various western agencies in the adoption of typewriters:

From my experience in civilian agencies here in the west, it's all over the place - the Denver Mint chugged along handwriting nearly everything into the first decades of the 20th century while the Bureau of Reclamation, stood up in 1903, was typewriting everything right off the bat. And it often depends on the type of document - for some, statistical forms and reports are often typewritten while general correspondence was still handwritten. Within just Bureau of Indian Affairs offices it's wildly different - just in Montana the Fort Belknap Agency was typewriting most everything by the turn of the century while the nearby Crow Agency was still handwriting some official correspondence in 1910.

Even within the Patent Office itself there was some inconsistency:

In the patent case files the Patent Office starts replying to inventors with typed letters in the late 1880s.  I've seen early typed documents from inventors in the early 1880s and maybe even the late 1870s.  The Patent Office wasn't consistent and you still see written responses after the first typewriter was used.

The History of the Government Printing Office, founded in 1861, is very informative, as it includes a history of the use of the printing press in the colonies and early U.S. government.

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