Can you play guitar without playing chords?
You could even rank chords above scales in importance on guitar, because you can play chords to accompany singing. A lot of musicians don't do much more than play chords on the guitar for accompaniment. You might find particular songs that don't use chords, but you won't find a guitarist that does not play chords.
Do you need chords in a song?
As songwriters, the chords you use and the way they're put together is important. It can help create the melody or give context to how the melody makes us feel. You can have as many chords as you like or work with just two – a progression has to have at least two. But a song can work with even a single chord.