It is correct and usual, at least in my variety of Spanish (Spain). The preposition 'a' is used with weekdays and dates, as in these examples:
Hoy estamos a lunes.
Hoy estamos a 19 de diciembre.
So, in order to turn these statements into question form, we have to replace 'lunes' with 'qué día' or '19 de diciembre' with 'qué fecha':
¿Hoy estamos a qué día (de la semana)?
¿Hoy estamos a qué fecha?
But the more natural word order in Spanish questions requires moving the interrogative expression to the front of the sentence, which gives us the following more common questions:
¿A qué día (de la semana) estamos hoy?
¿A qué fecha estamos hoy?
On top of that, the word 'día'/'fecha' is often omitted, resulting in the very common question:
¿A qué estamos (hoy)?
This is the way I would typically ask the question if, for example, I'm filling out a form in front of somebody and I'm unsure about the date.