Staying less than 3 months
If you are an EEA/EFTA citizen who intends to work in Iceland for less than 3 months you need to contact Iceland Revenue and Customs (Skatturinn), regarding the application of a system ID number. You can find the relevant form RSK 3.30 Application of a system ID number from Iceland Revenue and Customs. A tax card is issued following the allocation of a system ID number. You cannot apply for a system ID number yourself but you do need to request Iceland Revenue and Customs to apply for you. A system ID number does not grant you any rights in Iceland and does not confirm your right to residency in the country.
If you need the system ID number for other reasons than employment you need to contact the relevant public authority that requires an Icelandic ID number for their own purposes.
About System ID number