I doubt there were enough Icelanders to do this effectively. Iceland's population in 1940 was about 120,000; if it mobilized at a similar rate to the UK, that would give about 8,000 men under arms (compare 3 million out of 46 million for Britain, which is about 6.5% of the population). Typical destroyers at the time had 200-400 crew, but you always need folks at home - navies usually have pretty "bad" tooth-tail ratios, so we're looking at maybe a quarter or so on boats (which is a lot). So...5-10 destroyers. And frankly, that's being generous. And how many are at sea at any time? There's no way that Iceland can support a navy large enough to defend its own waters.
The only way you get an Icelandic Navy would be to re-flag some US, British, and Canadian ships with Iceland's flag and keep their old crews with a few token Icelanders onboard to help with local navigation and give the sham some realism.