We do not recommend that you use chemicals to attempt to unclog yard drains or driveway drains as they typically drain directly into a waterway via the storm sewer without first treating the water at a sewage treatment facility.
Fix or replace broken yard drain grates. Consider finer screen or grate to keep more dirt and debris out, but not too fine that will restrict the flow of water or easily get clogged up.
Be careful putting anything rigid into your drains as the drainpipe very well could be clay pipe and might be damaged by sharp or heavy objects.
If water still drains slowly after a thorough cleaning, consider having a larger yard drain installed. This could be costly, but it may be the only way to allow the water to drain quickly enough that it does not stand in your yard.
Once you find a drain cleanout or cleanouts in your yard, hand sketch a map of where all the drains and drain cleanouts are for future reference.