You don't have to be resident in the US to get an Amex. It's available in a lot of countries. Check with your local bank/s.
But if you want a US-issued Amex, you need to either be a high profile person who spends a lot of time in the US and/or build up a US credit history. A US bank account isn't strictly required, but can increase your chances. Banks in Wyoming and Nevada are not more open to non-residents. Go to New York, Miami, or LA.