Hi,
I don’t know of anyone who will rent a hiking carrier. I am hoping one of the camping gear rental companies will start carrying them; you could perhaps reach out to them and see if they are willing to start renting them:
https://www.iceland-camping-equipment.com/
http://outfitters.is/en/
As for strollers, there are several options for renting (or buying!), but none of them are super easy. If you can bring one, that’s your best bet; some airlines do allow you to gate check a baby item for free.
Otherwise, here are a few ways to rent a stroller:
The store Olavia and Oliver is an upscale children’s store that may rent strollers. But this is a store where parents in Reykjavik would shop for expensive children’s gear, and there may be a waiting list to rent strollers. They don’t show strollers right now on their rental page.
Here is a place that appears to rent strollers, and will bring them to your hotel. But I’ve never heard of them, and it looks like the web site was just created a few weeks ago. They might be great, but I just don’t know.
Since we were in Iceland a long time, we actually ended up buying an umbrella stroller from a store called Rúmfatalagerinn. Here’s a link to the stroller.
So it’s about $50; if you add more items to your purchase (on the same receipt) to get to $60 (6,000 krona) you should be able to get a tax refund in the airport. Just make sure they are items you will take home with you, and save your receipt!