Is it possible to wait for some cards to leave the stack to use them for delve?
Yes, you can. After each spell resolves but before the next one on the stack, both players get priority. You can cast more instants or activate more abilities then. Become Immense is an instant, and you are able to pay the mana costs, so you can put it on the stack.

A simpler way to look at this particular example is, you can cast Giant Growth (resolves), then Giant Growth (resolves), then Giant Growth (resolves), then Giant Growth (resolves), and with the stack empty, cast Become Immense.