After doing more research and experiment I was able to figure it out. The examples confused me (I think theyassume a lot of context about Azure's SOAPy conventions over REST, such as case sensitivity and quoted strings). Perhaps they will confuse others so I'm posting my answer here:

function searchBing() {
    var request=require('request');
    var url="https://api.datamarket.azure.com/Bing/Search/v1/Web?Query='sushi'&$format=JSON";
    var key="[your account key]";
    request.get(url, {auth: { user: key, password: key} }, function (error, result) {
        console.log(error, result.body);
    })
}