It is rather easy to establish a business in Iceland and almost anyone can do so. You mainly need to choose whether your company is going to be a Business Under Sole Proprietorship, Private Limited Company, a Public Limited Company, Partnership or  Branches of Foreign Companies. It depends on the business and sectors what kind of licence you will need but all businesses require a VAT ID number and you need to fill out an estimated payroll form. Note that all the forms are always in Icelandic so you will need some help from an Icelander with this. 

Business Under Sole Proprietorship are companies that an individual owns and runs under his/hers own personal ID number. These are often self-employed persons and the company holds their name. Taxes and salaries etc. are paid just like any other company. Link to the form https://www.rsk.is/media/rsk05/rsk_0502.is.pdf

Private Limited Company can have one or more shareholders and the share capital requirements must be 500,000 ISK or more including the 131,000 registration fee. When establishing a Private Limited Company a VAT ID/Payroll form must be submitted along with Company Charter, Board resolutions as well as notification of a PLC. According to the EEA agreement at least one of the founders as well as the board’s majority need to be domiciled in Iceland or within the EU or an OECD country. The Ministry of Industries and Innovation can grant exceptions. Link to the form https://www.rsk.is/fyrirtaekjaskra/stofnun-felaga/einkahlutafelog

Public Limited Company is similar to Private Limited Company but the initial share capital needs to be higher or be 4 million ISK including the registration fee which is 256,500 ISK. Applicants need to fill out various forms such as Company Charter, board resolutions as well as notification of the establishment and of course VAT ID/Payroll form. The company needs to have at least two or more shareholders with a Board of Directors with more than two board members. Here are more information and forms https://www.rsk.is/fyrirtaekjaskra/stofnun-felaga/hlutafelog/. Just like with Private Limited Company there are restrictions in regards to ownership’s nationality. 

Partnership is when two or more people share equal and unlimited responsibilities towards the liabilities of a company. Registration and fee varies depending on the agreement and whether there is a Board of Directors etc. Forms are in Icelandic https://www.rsk.is/fyrirtaekjaskra/stofnun-felaga-i-atvinnurekstri/sameignarfelog/  but there are no conditions in regards to residency of owners.

Branches of Foreign Companies require various documents to be filled out and registered branch must have a name that included the name of the foreign company. The registration fee is 256,600 ISK.