You'd think so. Maybe they could. Maybe they should. I too think a reference number or something back would be expected, but they don't. They don't tell you that you'll get a receipt confirmation, but they do tell you they will contact you with the result. If I hadn't heard from them, I'd be chasing them, because they make it pretty clear what happens otherwise.
Good
business practice and common sense would have a replay email acknowledging receipt of her email with information on what the next step is including words to the effect "
if you do not receive conformation within ## hours, please call ## #### ####"