This would seem a logistical problem that should be addressed for future storms. Perhaps someone could arrange to have some vehicles full of cages that could be taken somewhere for some fee. Unfortunately, some do-gooder would probably take legal action against any company that tried to charge a fee even remotely high enough to make such an effort worthwhile (charging "price gouging"), though even a $2,500 fee (including pre-paid boarding for a small animal up to three months) would probably have a lot of takers.
Your idea is a very good one and is already done in some areas down here.
In the Redlands, farm land in southern S. Dade, their are land owners who have built hurricane "proof" stables for people to board their horses in the event of a hurricane.
Also some of the vets here allow people who are leaving to board their pets at the vet facility for a reduced cost. Obviously their space is limited, and thereis no control over the workers at the office being able to care for the animals in the event of a major storm.
If I knew of a company such as the one you suggested, i'd be willing to use them if it meant my animals would be safe.