WASHINGTON—As the partial government shutdown stretches on, some House Democrats have grown concerned by what they see as the absence of a clear strategy from party leaders over how to end it.

Some newly elected Democrats are increasingly frustrated that they are taking complaints from angry constituents without having a mapped-out plan to end the partial shutdown—which became the longest in modern history over the weekend. Many believe they were elected to Congress to try to end its dysfunction and had hoped to be reaching...