How many spokes can break before I need a new wheel? 3 of the spokes on my wheel are broken. How can I tell if the wheel is still good to use?

Can I just replace the spokes or do I need to buy a brand new wheel?
The spokes support the wheel under compressive load. The rim in itself is relatively weak in compression. Of course, losing a spoke means that the remaining spokes have to carry increased load and you risk breaking more spokes or buckling the rim beyond repair.

(Note: minor buckles can be removed through "truing" the wheel, a process of adjusting spoke tension to pull the rim in both the axial and radial directions so it is round and straight again)

Recommendation:

Have a bike shop assess whether the rim has been damaged from riding on 3 less spokes. If they deem it is still usable, then they can replace your spokes (possible more than just the 3 which broke) and true the wheel if necessary.

Further reading:

There are some interesting websites that consider analysis of a spoked wheel:

Spoked wheel analysis

Bicycle wheel stiffness