What factor has the strongest influence on supply in an economy that grows rapidly?
The best answer to this is that the availability of resources is the major determinant of supply in an economy that is growing rapidly. This is true both for the supply of individual goods and for aggregate supply.
Resource availability is not the only important factor. For example, technological growth is a very important determinant of supply. However, this is true for all sorts of economies so it is hard to identify it specifically with rapidly-growing economies.
Resource availability is the most important factor because there will be a great deal of competition for those resources in an economy that is growing rapidly. For example, labor is an important resource. In an economy that is growing quickly, workers are in high demand. There may not be enough good workers to go around. This will cause supply to fall. However, if something were to happen to increase the workforce (for example, increased immigration) various firms would have enough workers to makes all the products they would like to make. The same dynamic is at work for other resources such as fuel or iron.
In a rapidly-growing economy, there is a great deal of competition for resources. The availability of resources is the main factor that affects supply in such situations.