Why didn't the states of the Arabian peninsula industrialize around the time that Europe and East Asia did? Was it simply a lack of accessible natural resources? Why not import coal?
Europe and East Asia didn't industrialize at the same time. In all of East Asia, probably the first country to Industrialize was Japan, and that didn't happen until the right around the beginning of the 20th century, nearly a century after the process started in Western Europe.
So really you have to compare with Europe. The big advantage here was simply one of demographics; both sheer numbers and growth.
In 1815 all the Arab states together had about 13 million people. There were perhaps about 6 million more Arabs living in the Ottoman empire (The empire had a respectable 25 million, but most of those were Turks or Balkan Europeans). This is barely more people than were living in the same area 500 years earlier. Supplying goods to that cultural area in the same way they were supplied in the 1300's was probably not much of a strain.
In that same period of time, Europe's population (not counting Russia, which is kind of a special case) exploded from about 67 million (roughly 5 times the Arab states) to nearly 200 million. Keeping 200 million human beings clothed, heated, and fed every year simply required new more efficient methods of weaving, heating, and farming.
Saudi Arabia encompasses large deserts, and industrial goods have a low "value to weight" ratio. The goods that are likely to be shipped across such deserts have a high value relative to their weight, such as cinnamin, silk, and precious metals.
At the time of the Industrial Revolution (in Europe), most people of Saudi Arabia were still nomadic. People who are nomads are not likely to build factories to "industrialize."
Also, at the time of the industrial revolution, the Hejaz, or the most advanced parts of Saudi Arabia along the Red Sea, was directly under Ottoman rule, rather than being part of the "vassal" state known as Saudi Arabia.
Only in modern times has Saudi Arabia had a commodity, oil, that makes it worth industrializing.
Imperialism mainly. Egypt's attempt under Mehmet Ali was crushed by Britain and France, and the Ottoman Empire was saddled with 'free trade' treaties and concessions that made even the first stages of industrialization impossible. Other Arab regions were either colonized early on (e.g. Algeria and Tunisia) or were too impoverished and sparsely-populated (e.g. the Arabian Peninsula).
By the end of WWI nearly all Arab countries were occupied by Western powers, who installed elites that had little interest in industrialization. The military regimes that overthrew them in places like Egypt and Syria tried to make reforms, particularly agrarian reforms, but the wars with Israel and Cold War geopolitics generally frustrated their efforts at industrialization. Armies then became vehicles for vicious sectarian or tribal dictatorships. After that came the neoliberal era, where regimes ceased to even give lip service to economic development.
During that time, Saudi Arabia used its enormous oil revenues to build a modern infrastructure with generous social welfare programs, but its efforts at industrialization were very limited and frustrated by the 'resource curse' which made its currency too strong and its labour uncompetitive.
Afaf Lutfi El-Sayyid Marsot's A Short History of Modern Egypt offers a good treatment of Egypt and the Ottoman Empire in particular.