They have no garbage service? As in, they have no receptacles? If it’s a restaurant, that might be a matter for the health department. It seems like they would be required to have sanitation.
If you live in an apartment building, and pay a fixed amount for water/sewer/garbage, then overflowing the dumpsters isn’t your problem. Any overages should be paid by your landlord unless you have another written agreement. If the landlord gets upset about continually paying extra, then they should start caring that another building is using their dumpster. If you rent a house, can you store your trash cans where they aren’t accessible? Dumping trash is definitely illegal, so perhaps some calls to the police are in order.