This song was written in 1909, an era of songwriting that seems nostalgic. Many of the lyrics of that time use the same rhymes, over and over again. They liked to talk about lovers courting at night, by the light of the moon, and often featured a set of words that rhymed with "moon."

As a clue, I'm going to list a number of words that rhyme with "moon." Some of them appear in the song and some do not. This list does not include all of the puzzle words.

LIST OF CLUES:

A boon is a benefit. "The trade winds were a boon to sailors in the age of sail."

To croon is to sing quietly. "In the old days, singers needed to project their voices to be heard in an auditorium. But in the 1930s and 1940s, singers discovered that they could to croon into a microphone."

A goon is a rough person who has been hired to beat someone up or scare them. 

June is the sixth month of the year.

A loon is a large diving bird, famous for a mysterious cry like a person laughing.

Noon is the middle of the day, 12:00 hours.

To spoon is an outdated expression from a century ago, meaning courtship behavior like hugging, kissing, and perhaps sometimes more.

Each underbar is to be replaced by a single letter. I am showing the rhyming parts of each word.

THE PUZZLE:

By the light of the silvery _oon,

I want to _ _oon, to my honey I'll _ _oon love's _une,

Honeymoon keep a-shining in _une,

Your silv'ry beams will bring love dreams, we'll be cuddling _oon,

By the silvery moon.

I've chosen a YouTube video sung by a "barbershop quartet." Barbershop is an American folk style, originally developed by African-American singers, then revived by white singers in the late 1930s. It is sung by an unaccompanied male quartet, singing very precisely tuned chords that create a "ringing" quality. Barbershop emphasizes songs from the early 1900s that contain simple chords that "ring" well.

The song begins at about 00:09.

https://www.youtube.com/watch?v=IOHpLP8OUA0