How can I manipulate the vertices of the sphere in three.js?

It’s a little hard to answer this without knowing exactly what you want to do.

However, a common approach to this problem is to create a new object (say, a new tiny sphere) centered on each vertex of the big sphere.

Can you explain in more detail what you want achieve? It will also help if you share your code too. There’s some templates here that you can use to get started: