A Vermont mead brewery has delayed a major expansion because the owners can’t get a business loan. A craft-burger joint in Utah has sent employees home because sales are down. And a Maryland Lyft driver has seen his weekly income fall by nearly half.

The partial federal government shutdown that became the longest on record Sunday is curtailing infrastructure projects, food-processing inspections and economic data used by Wall Street. But on a more micro level, it is showing signs of disrupting commerce as hundreds of thousands...