I need a database that contains a complete list of commercial restriction enzymes: name of the enzyme & recognition site.
You are probably talking about type II endonucleases, there are a few databases available from the companies which sell these enzymes like NEB or Thermo Fisher. Additionally tools like NEBcutter, where you can paste a sequence and get all recognition sites for restriction endonucleases in this sequence are useful. I linked them below in the references.
Edit: Thanks to @Michael_A I have added the lonk to the Rebase Database. It is a database which tries to be a complete ressource on restriction enzymes.