Where or how is the best way or place to get dollars? I want $4500 dollars to take on vacation to Florida with me and rather than get them in the UK when I go home I was hoping it would be cheaper to get them out here as the exchange rate for sar to dollars is a lot more stable than sterling.
You are right it is "cheaper" to exchange the SAR but not for the reasons you are thinking.
The SAR exchange rate to USD is fixed/pegged at 3.75. Since USD/GBP ("cable") is floating that is the only volatility in the GBP/SAR rate - ie GBP/USD falls 1% therefore GBP/SAR falls 1% and so on.
If your choice is between selling GBP or USD to buy SAR then you should sell the SAR since you are not really having to make a market transaction. Banks in the region take a small spread on the USD/SAR rate, but substantially less than a tourist exchange rate would on GBP to USD rate.
And, I agree with GroveWanderer that any bank in the region will make the switch for you.
The banking rules have changed here since I posted in April 2015. The banks now require you to have an account to do most anything with them. It was a Saudi Banking law that is now being enforced by the Ministry and the banks on the Aramco camps are having to follow.