お～願う【お～ねがう】

お～願う is a formal request, "I must ask you to ~". The verb goes in the ます-stem between お and ねがう願う, and 願う almost never appears in the plain form: you meet it as ねがいます願います, as ねがえます願えますか for a question, or as ねがいたい願いたい in writing. Announcements, notices and business letters use it.

As with おから～ください, there is no て: ×お待って願います.

With a noun

A Sino-Japanese noun that names an action takes ご in place of お and goes straight before 願います, with no を. This is the form on signs and in emails.

A few な-adjectives work the same way with に. おしずかに静かにねがいます願います is the one you will hear.

お～願います or お～ください

Both are honorific requests. おから～ください is the everyday one at a shop counter. お～願います is stiffer, and in speech it can sound like an instruction you are not free to refuse; おひきとり引き取りねがいます願います is how you tell an unwanted visitor to leave. The question form 願えますか is the soft version.

お願いします

おねがいしますお願いします is the same verb in a different shape: the thing you ask for takes を and comes first. おねがいいたしますお願いいたします is humbler, and おねがいもうしあげますお願い申し上げます belongs to letters and formal speeches.