Well, English doesn't have a standard way of framing such a question. But there are several possible ways:

• Where do you come in your family?
• What child are you in your family?
• Where among your siblings do you come in order of age?
• What is your relative position in terms of birth order in your family?
• Among your siblings, where do you fall with respect to birth order?
• What is your ordinal position among your siblings?
• What is your natal order among your siblings?
• What is your birth order?

Notice that in most cases you will get answers like "youngest, oldest, middle and so on." To avoid that I still suggest you include numbers in your question:

• What child are you in your family? First, second, third?

On the other hand a simple "Are you the youngest, the middle child, or the oldest?" will get you straight what you need.

There is no standard way of asking this without using an ordinal or the words "youngest", "eldest"/"oldest" in the question.

("Where do you fall in the birth order among the siblings?" would get you some puzzled looks.)

A native speaker might ask, "Are you the youngest, eldest, or ...?" or maybe "Are you the eldest, second, third, or what?" But neither of these solutions would abide by your rule of avoiding numbers.

You could try, "How many of your siblings are older than you?"