Do invertebrates have a respiratory system?

Yes, Invertebrates have respiratory system.

upto phylum Annelida no special respiratory organs are found. The respiratory system first appears in phylum Arthropoda. Trachea, Book lungs, book gills are respiratory organs in Arthropodans.

Mollusca have gills.

Echinodermata have Tube feet, Respiratoryy tree and bursae