Yorkshire won a total of 11 medals made up of 8 gold, 1 silver and 2 bronze. With a population of 5.3 million that would give them a per capita medal count of about 482 thousand per medal, which would put them sixth in the total medals per capita table. However there is a problem with the 8 golds figure which is glossed over in the news media - only 4 were won by individuals and the other 4 were in team events. Teams representing the UK are very unlikely to be made up entirely of people from Yorkshire.
The news story about Yorkshire's medal winning performance illustrates the main problem with the whole per capita medals idea in that it could also be applied to regions within countries or to cities in countries. I can remember New Zealand pushing this per capita medals idea back in the 1980s as they tend to do pretty well when the statistics are presented this way, and I'm not surprised that the medalspercapita website is New Zealand based.
London actually did better than Yorkshire in the Olympics, winning a total of 12 medals (8 gold, 2 silver and 2 bronze, with 3 of the golds in team events). So I think the fact that Yorkshire's good performance is much more well-known than London's is down to Yorkshire's not insignificant ability at self-promotion.
Yorkshire actually don't promote themselves nowadays as much as they did a few decades ago. The self-promotion was probably at its height in the 1970s, sparked off by the Yorkshire born prime minister Harold Wilson, who pretty much invented the idea of the "professional Yorkshireman" taken up by various others including Michael Parkinson, Freddie Truman, Geoff Boycott, Arthur Scargill, etc.