There are numerous ways musicians learn songs by ear. Some methods are faster and more efficient, others are time consuming but more accurate. This method that I follow can be a little time consuming. But following this method, not only you can find out the chords but you will know a lot more about the structure of the song, which will allow you to be more accurate with your chords. This will also give you the control to modify and perhaps improvise to give your own touch. More importantly, this will allow you to find out chords for songs where no guitar has been used. Following are the steps:

Step 1. Follow the vocal lines and try to hum the essential notes.

Step 2. Find out how to play the notes of the vocal lines, one line at a time.

Step 3. After playing a few lines on guitar, try to understand the tonal centre, where all the notes want to resolve into, or lay out the notes of the vocal lines you played and see if they form any scale pattern.

Step 4. Once you identified the key of the song, find out what diatonic chords are available for that key.

Step 5. Use trial and error to find out the chords that matches the actual song. Most chords are going to be diatonic chords, occasionally you may find one non diatonic chord here and there but that is very uncommon in popular music.

Step 6. Find out the strumming pattern. If there is no guitar in the song, create a strumming pattern that matches  the drums and bass grooves.