Anyone who is legally resident in Denmark can set up a business. You should have your own personal CPR number and when you register your business, it will have its own separate number called a CVR.
It is easy and quick to set up a business in Denmark. You should be careful that you understand your responsibilities when you have set up a business. For example, you must report VAT (MOMs) even if it is to say that you have not sold or bought anything with VAT in the last 3 months. If you do not report, then you will get fined about 800 DKK. This is just one example. This is why it is a good idea to pay for an accountant who will take care of all this for you.