Luckily, I found just such a store conveniently open on Saturday afternoon called Lou Lou Buttons (212-398-5498). They're located at 69 East 38th Street right near 6th Avenue and they're open from 10am to 6pm on Monday through Friday and from 11am to 5pm on Saturday. They have loads of buttons! It's a slender little drawer of a shop filled with nothing but buttons. Row after row after row of all sorts of magical little shapes in every color and size you can possibly imagine. It's neatly organized and the guys working there were super friendly.
I showed the salesman exactly what I needed and in less than 5 minutes he expertly pulled out a few boxes of buttons until he found the exact buttons I needed. He filled them into two little plastic bags and I was on my way. Now I just need to find a girl who will sew them all on for me.