As already said previously, make sure that your card belongs to an international network such as Visa or Maestro (debit). There are very few occasions in which card payment is not possible.
Electronic payment is also possible on board ferries.
It is normal to buy a cup of coffee with a card in Iceland. But since you'll be using a foreign card keep in mind that exchange rates apply, plus there may be a fixed commission on every payment. So if you buy a low-value item like a coffee, the effective end price could be multiplied by two on your credit card statement due to the service fees.
So for very small purchases (< 1000 Kr) I tend to favor cash when I have it.
Visa and Mastercard + Maestro have widespread acceptance, Amex is accepted in many outlets too, but not as much.
Discover, JCB will probably be accepted in hotels, restaurants, tourist shops mostly.
There is broadband Internet almost everywhere, unless you venture into the highlands which is a desert.
The lack of cash can also be explained by the fact that the practice of tipping is rare in Iceland. You are normally not expected to tip even for good service but it's not forbidden. Therefore cash is seldom needed.