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

If you get started you might learn from examples in this forum.

You may need to mix some of the techniques from the examples.

Some examples partly to the sphere.

https://hofk.de/main/discourse.threejs/2017/PictureBall/PictureBall.html

https://hofk.de/main/discourse.threejs/2018/LabelsOnSphere/LabelsOnSphere.html
https://hofk.de/main/discourse.threejs/2018/SetRemoveMarkers/SetRemoveMarkers.html
https://hofk.de/main/discourse.threejs/2018/Paris-lat-lon/Paris-lat-lon.html
https://hofk.de/main/discourse.threejs/2018/Tree%20on%20a%20Sphere/Tree%20on%20a%20Sphere.html
https://hofk.de/main/discourse.threejs/2018/SelectSegments/SelectSegments.html

https://hofk.de/main/discourse.threejs/2019/MultipleImpactsEffect/MultipleImpactsEffect.html
https://hofk.de/main/discourse.threejs/2019/SelectFacesWithinRadius/SelectFacesWithinRadius.html
https://hofk.de/main/discourse.threejs/2019/ConstructionBasis/ConstructionBasis.html

They could be useful for your purpose.