Do I need a certificate to teach Yin Yoga?

No, you do not need a certificate to teach "yin" yoga. Yin is an adjective, not a noun. If you believe the style you teach is best described as "yin" you do not need anyone's permission to call it that.

Although the vast majority of yoga taught today is yang in nature most of the yoga books published before 1985 were advocating what we would call a "yin" style of practice. The yang styles of yoga are what's new, not the yin style. The yang styles are great, I am glad we have them, but sooner or later everyone finds a yin and a yang in their own pratice. This is inevitable because all things have a yin and a yang aspect, yoga is no exception. I have met many, many students and teachers who have confided that they have "Always practiced yoga this way - at home - when no one was looking - but I didn't have a name for it."