The Icelandic Pony (aka Icelandic Horse) didn't start out as it's own breed, but resulted from a mixture or cross-breeding of different types of horses / ponies.
The Icelandic Horse (Icelandic Pony) is, most likely, a product of a combination of several European pony breeds. Tracing the exact ancestoral lines of the Icelandic Pony may not be entirely possible, but we can assume that the original stock were probably from the British Isles and Norway.
The British Isles and Norway have several breeds of ponies including: Connemara Pony, Dales Pony, Dartmoor Pony, Eriskay Pony, Exmoor Pony, Fell Pony, New Forest Pony, Shetland Pony, Welsh Pony, Highland Pony, Nordland (aka Lyngshest) Pony, and Fjord Pony.
The pony breeds from these areas are fairly similar, being short and stocky, acclimated for colder temperatures, with short extremeties (including ears) and heavy winter coats. Some of these pony breeds may be descendants of the Celtic Pony which was native to those areas at the time.
The Vikings colonized Iceland around 900 AD. The ponies that were taken with them had to be small enough to fit in their boats, which were open, to make it from one continent to another. A low center of gravity was probably a plus for the trip, and may have kept some horses from falling overboard.
Possibly the boats included the Fjord pony and a group of ponies from the Lotofen Islands. Additional later settlers from other areas such as Scotland, the Orkneys, or Shetland brought their own ponies.
Some resources indicate that there may have been Mongolian blood involved in the creation of the Icelandic Pony, along with Fjord and Tarpan. These have blended into one breed, but various types and sizes can still be seen. Some of the Icelandic Ponies are gaited; some are not.
There has been molecular biology research on the genetic relations between horse breeds in Norway. One research project dealt with four Norwegian horse breeds, the Dales, Fjord, Norwegian trotter and the Nordland / Lyngen; and included two British breeds, an English racehorse, an English Standardbred, a Shetland pony, and a Mongolian horse.
The comparison described above showed that the Shetland pony and the Icelandic horse are closely related, and both more related to the Nordland / Lyngen horse than to any breed in the study.
For the past 1000 years, the resultant pony has been isolated on the island from any further mixed breeding.
There was quite a limited gene pool, much more so since the 1783 volcanic eruption that killed approximately 10,000 people and about 3/4 of the island's livestock, in addition to devastating the vegetation. Deaths were caused by volcanic haze (cloud of volcanic gases and particles), contaminated vegetation and water, and famine.
As the grasses withered, and then started to recover, much of the livestock would not even eat the new growth, actually dying of starvation. Some were poisoned by the fluorine, some may have gotten "tedra teeth" which is brittleness caused by the ash, resulting in inability to effectively masticate. Heavy ash can also lead to bone development problems.
The most moderate calculation indicates a number of 19,488 horse deaths due to the volcanic eruption. That would have left about 7,000, throughout the island, therefore the last 200 years have had little opportunity for expanding the genetic pool. Considering the pocketed areas of settlements, proximation would have further limited the gene pool.