Do I have to pay this cancellation fee?

From the way you described it, it looks like you agreed to a contract with specific terms, which include cancellation penalty. Unless the contract stated the penalty would only apply once you made your first payment, then you owe the cancellation fees, and thus yes, you are responsible to pay it. Paying them would of course be your first option.

Your second option is to negotiate. If the reason for your cancellation is the recent change in law (which happened AFTER you signed the contract), you may have some recourse. In this case you should reach out to them, and explain the situation. You would still likely have to pay some cancellation fee, but this way you can have it reduced.

And of course your third option is not to pay. If you did not give them the credit card, and they do not have physical presence in your country, their chance to collect it are rather low. This however does not invalidate the debt - for example, one possibility is that they can sue you in Iceland in your absence (if in your contract you agreed to Icelandic jurisdiction), and get the default judgment. This may be enforceable in your country, and would certainly apply if you ever visit Iceland again. Thus attempting to resolve it peacefully with both parties satisfied may be a better strategy in the long run.

This depends on the assumption that the contract you've agreed to contain the cancellation provision, requiring you to pay 75% of the total amount in case of cancellation. If it contain nothing, and the company just made up a number, the situation of course is different.

Contracts are binding.

You accepted a contract, and no matter how outrageous the conditions are, you are bound by it. You cannot apply the american 'feel' that the law protects you from 'surprising' fine print; other countries have other laws, and especially this protection is mostly non-existent in Other parts of the world. You are expected to read what you sign before you sign.

In addition, consider their position - this is just an example, of course - imagine they had no motorhome available for your booking period, and now shipped one to iceland just to rent it to you. Now you don't want it, and they sit on the cost.

Many of the other answers and comments are operating under the assumption that a binding contract has been made; it is not at all clear to me that this is correct. Some of the moralizing seems particularly uncalled for.

Typically (western common-law based) courts require some sort of reasonably tangible consideration to change hands on both sides in order for a contract to exist at all; clarifying this point is the reason why things are often sold for a dollar (or other token sum) rather than simply being given away.

In this case, I can't see where either party has received anything of value; the rental agent has received nothing from you (you did not pay the invoice), and it seems unlikely that a judge would consider that reserving an RV for you two months from now is of any value. (Given that this is an easily reversible decision)

Of course it is possible that Iceland has some sort of unique Viking vacation rental laws that you could run afoul of, but barring this I would strongly suspect that the rental agent is fishing for a sucker, and you should not pay.