Wow. I'm certainly sorry for your trouble with this - - I'm sure it is not pleasant for you!
So, I assume that even if the storm drain weren't filled/blocked, you would still have run-off issues since there's nothing to stop gravity from dumping sediment and whatever else is on the neighbor's property onto yours.
In that case, you certainly can go to the board of the HOA and ask that the neighbor be required to put up a retaining wall between your property lines to prevent this.
IF the HOA or neighbor refuses this, then you would have the right to sue for a private nuisance and seek a permanent injunction against the neighbor, which would be a court order preventing/prohibiting the neighbor from allowing debris, dirt, water etc. from his yard to dump onto yours, causing flooding, damage, erosion, etc.
This would effectively require the neighbor to construct something to prevent runoff of his yard from destroying yours.
You should consider speaking to the HOA and neighbor first to see if you can agree to erect a retaining wall, etc. to stop the problem. If they aren't agreeable to make this happen, then your only recourse is to sue and claim a private nuisance that is destroying your land and also preventing you from your right to quiet enjoyment and ask the judge to give you a permanent injunction ordering the neighbor to keep any debris, runoff, water, etc. from eroding your property.