We live in Vancouver, Washington, where it rains most of the winter. Our downspouts go down the side of the house and disappear into a hole in the ground. However, there is a clog at some point underground. I don't know if they eventually connect up to sewer or storm sewer.
The house was built 27 years ago. We bought it 4 years ago, but gutters were already clogged. We have 4 very large Douglas fir trees along the back of the house that provide plenty of stuff to clog the gutters all year long. I clean the gutters 2 to 3 times a year, and in the first year we owned the house I added wire mesh thingies over the places where the downspouts connect to the gutters. That keeps more stuff from going down, but that is like closing the barn door after the horses got out.
What is the best way to solve the problem? Is it renting a rotor-rooter type machine? Is it giving up on the undergound system and diverting the water on the surface? A previous owner put a lot of large concrete areas around the house, so water doesn't get into the crawl space.
Based on your description, it sounds like the home has a history of crawl space flooding, which very well may be caused by clogged underground drain pipes.
I would first evaluate them to see if they can be saved. Hire a drain cleaning company that can snake and clean the pipes and run a drain camera down the lines so you can see exactly what's going on. If the lines are crushed and not savable, then you can make the decision whether or not to reinstall an underground drain system. Your other option is to make sure the gutters and downspouts are extending out away from the house and not going into underground drains.
It's very important to know where the water is going. If it's backing up underground, it could be damaging the foundation.