What would you say in everyday life if you wanted to buy them?
We need to buy some ______.
- pen ink holders
- ink reservoirs
- pen shanks
- pen rods
- small tubes with ink?
According to howstuffworks.com, How Ballpoint Pens Work, it is called an ink resevoir:
The ball rotates freely and rolls out the ink as it is continuously fed from the ink reservoir (usually a narrow plastic tube filled with ink).
However, in a context where you need to buy more and you're at the store, I think that you should ask for pen refills. From Office Depot: