What it means
を願う follows a noun phrase marked by を and expresses wishing for, hoping for, or requesting that thing. In formal notices and official statements, it commonly makes a respectful request for cooperation, support, understanding, or a response. The plain form is especially characteristic of formal written language, while を願います is the natural polite form for addressing people directly.
Usage note
As a standalone request, plain を願う can sound formal and somewhat detached; it is most natural in official writing or when the subject is stated as an organization. Use を願います for a direct polite request. This construction can also express a wish or hope, so its exact meaning depends on the noun phrase and context.
How to use it
事務局は、皆様のご協力を願います。
The office requests everyone's cooperation.
事務局は、皆様のご協力を願う。
The office requests everyone's cooperation.