Weak mudsill aka "Sill Plate" causes stucco failure.

My husband and I recently bought a small Southern California bungalow built in 1938. The stucco is crumbling on the bottom half of the N and S sides of the house - it is at the same height as the crawl spaces. The home inspector thought it was a bad stucco repair job with some water damage.

We had a contractor out to bid on epoxying two small cracks in the foundation. He said that our stucco is coming off due to the ventilation holes being cut through the mudsill, and that we should replace the entire mudsill replaced and rebolted before having the exterior redone. Does this make sense to you?

Yes, this makes sense that cutting the mudsill, known as the sill plate here in Virginia, can cause the stucco to fail. Even if it isn't the main reason for the stucco to fail, the sill plate should be replaced, patched or corrected before the new stucco goes on.
The sill plate is the horizontal 2 by 4, or whatever framing lumber, that goes down against the foundation. The sill plate holds up the weight of the entire house. Even a small amount of movement in the sill plate can cause major damage to the stucco. Rotted sill plates are a large cause of our repair work, where the stucco is loose, or pregnant, at the bottom.
Anchoring the mudsill for prevention of damage due to earthquakes, is particularly important in your part of the world. There are anchors made for this reason. In an earthquake, the house can shake itself off the foundation. Anchor bolts on the foundation holding on the mudsill can prevent this.

Here, a rotted sill plate caused a wall to buckle and bow. There were other sorces of failure, such as interior lath used outside, but rot and compression of the sill plate caused the wall to bow:

Rotted sill plate caused stucco to bulge in Washington, DC.

Sill replaced, sheathing replaced at bottom with zip wall.

Mudsill replaced.

A Termite damage required a new sill plate and cripple studs. In this case the cripple studs were scabbed onto the old studs. Sometimes, a cripple wall is built under the old wall with new horizontal plates.

Sill plate replaced due to termite damage.

Another stucco patch job we did due to insects eating up the sill plate:

Another termite damage sill plate.