How do I achieve the following, making sure functionB() executes after functionA() finishes?

I think what you want is a third async function and then call that one. Like this:

function functionThatCannotHaveAsyncKeyword() {
    functionC()
    .then((resp) =>{
        // this code will execute after functioA and fucntionB finish. 
        // They will execute in that order thanks to functionC.
    });
}

async function functionC() {
    await functionA();
    await functionB();
}

async function functionA() {
    console.log('first');
    await someServiceThatMakesHTTPCall();
}

async function functionB() {
    console.log('second');
    await someServiceThatMakesHTTPCall();
}