Microsoft Word comes with an option (smart quotes, which is on by default) to automatically replace straight quotes ( " and ’ ) with left and right quotes ( “ ” and ‘ ’ ). I believe Photoshop also has such feature, and probably others have it too.
It is best to add those quotation marks to your font. In FontCreator select Insert from the main menu, and then click Insert Glyphs. From there on the first tab (By Name), enter:
quoteleft
quoteright
quotedblleft
quotedblright
This will add the missing glyphs along with their character code-points. All that is left, is to provide proper outlines. 