August spends its four weeks between flat and waist high. The first real depressions of the autumn are a September event, so the swell has not returned yet, and the month is defined by access rather than by waves: every road is open, the interior is reachable, and a wrong call about a spot costs you a drive and nothing else.
That makes August the honest answer to the question people actually type, which is whether you can surf Iceland in the summer. You can, it will be small, and you will still want a hood at 12°C. Watch the last week of the month: the first autumn lows sometimes arrive early, and almost nobody is there for them.

