La Capra Rampante has what you are looking for. it is in the Flaminio neighborhood and is located a 10-minute tram ride (followed by a short walk) from Piazza del Popolo http://www.lacaprarampante.com/
even non-organic goat milk & yogurt is difficult to find, Canestro has it and the farmers markets, too, and sometimes even the billa on via cola di rienzo but I am not aware (am not saying doesn't exist) of any organic versions.
Not Piazza di Spagna, but Canestro in Testaccio (I think via Luca della Robbia) is reliable. For local organic produce, the farmers market on via San Teodoro (near the Circus Maximus) is open on Saturday and Sunday.