We are a school district public library planning a capital project. The question is whether or not the project has to be approved by a public vote. We have been given money from our assemblyman towards the cost of the construction of one item in the plan. The remainder of the funds will be from the Friends of the Library, a foundation that is raising money in memory of two people and other private donations. We are not asking for tax dollars for the project.
There are a few scenarios where a capital project, such as a renovation, could start with a vote of the electorate of a school district library. If the project is subject to a bond, requires a tax levy increase, or is somehow tied to a referendum, the voters' go-ahead might be needed before work can begin. In addition, if a municipality or district was deeding over a gift of real property, that could require a public vote, as well.
However, in the scenario described by the member, the money is "in the bank" and is not conditioned on obtaining further funds from the taxpayers and no additional real property needs to be purchased or funds need to be levied or raised.
With that, while in library law I make it a rule to "never say never" in this case, I don't see a need for a public vote. Just follow the rules of procurement!
 Which is how the phrase "public vote" is used in the question.
 "Work" has a variety of meanings in this context; it could include hiring an architect, or purchasing real property, or putting an actual shovel in the ground.
 As with other major purchases by public libraries, compliance with competitive bidding requirements in capital projects is key, even if the library is using donated funds. For more on this, visit https://www.osc.state.ny.us/files/local-government/publications/pdf/seekingcompetition.pdf.