In reply to RLimmer:
There was a post about this a while ago. Try searching the forum. I went in April with my daughter on a non climbing trip. Even at that time of year there were climbable ice falls everywhere. You will need some sort of vehicle. We flew to an airport in the SW from Reykjavik and spent 5 days driving back along the coast and through the mountains. There are several hotels, but not sure if they are all open in the winter. Try emailing
https://www.mountainguides.is/ to see if they can give you further information. Many of the ice falls we saw were close to the coastal road, but if you venture along the valleys that run inland there are an almost unlimited number of possibilities.
Iceland in expensive and the weather very changeable. Violent storms can arrive quickly, with winds of 100mph not uncommon. However, weather is often very localized and by studying the forecasts you can usually move to an area with better weather. There is very little traffic once away from Reykjavik. I think the road round the coast is open all year, but many inland roads are closed all winter.