Step 1: Recall the concept of biological pump.
- The biological pump refers to the process where marine organisms fix atmospheric CO$_2$ and transport it into deeper ocean layers.
Step 2: Identify key organisms.
- Phytoplankton: Photosynthetic organisms; directly absorb CO$_2$ from the atmosphere and form the base of marine food chains.
- Zooplankton and fish: Consumers, they do not fix CO$_2$, instead they respire.
- Radiolarians: Protozoans that build silica shells, but not the primary fixers of CO$_2$.
Step 3: Conclusion.
Hence, phytoplankton are the key organisms responsible for CO$_2$ uptake in the biological pump.
Final Answer:
\[
\boxed{\text{(C) Phytoplankton}}
\]