Theres no requirement to be licensed with your council for carrying out Dermaplaning treatments. You will need a sharps box for your blades - this should ALL be covered in the training. Ensure you meet all the prerequisites for derma planing as my friend didn't and couldn't get insured until she did her level 3
If you require a sharps box for disposal of clinical waste you will need to contract with a clinical waste company to provide an audit. They will provide your sharps box, collect and dispose of it as per the contract. You will need to keep your collection/destruction certificate for 2 years just in case your EHO wants to see it. The clinical waste is your responsibility until you have the certificate of destruction from your statutory clinical waste company. You also need to carry and store it appropriately when not in use. This can cost as little as £30 per year for one collection. Depends on the company you go with. Shop around for quotes. As to licensing, you will have to contact your local licensing authority. Some require a license, some don't. They will also be able to advise you on the statutory clinical waste requirements for your licensure. Once that's sorted, and assuming you have a minimum of Level 2 in Beauty Therapy and your Dermaplaing qualification, the insurance should be a doddle. Some of our graduates have got it for about £65.
Hope this helps.