To be issued a machine operating license, you must have a valid driver's license, among other things. Therefore, it is not possible to acquire the right to operate heavy machinery without taking a driving test.

To be issued a machine operating license, you must hold a valid driving license for motor vehicles in Iceland.
It is therefore not possible to operate work machine in Iceland unless you have completed a driving test.
If you hold a valid driving license from another EEA country, it must first be converted to an Icelandic license before applying.