You want to run the loop while the value of `AMI_STATE` is equal to `pending`… so write just that.
while
AMI_STATE=$(aws ec2 describe-images --region "$REGION" --image-ids $US_EAST_AMI | jq -r .Images[].State) &&
[ "$AMI_STATE" = "pending" ]
do
sleep 10
done
case $AMI_STATE in
"") echo "Something went wrong: unable to retrieve AMI state";;
available) echo "Now the AMI is available in the $REGION region";;
failed) echo "The AMI has failed";;
*) echo "AMI in weird state: $AMI_STATE";;
esac