Learn Spanish (Latin America): El enfermero va a mover la camilla como usted pida. - The nurse is going to move the stretcher as you ask.

Because the action is in the future, so the exact way you will ask is still unknown. When como (meaning 'how' or 'in the way that') refers to a future or uncompleted action, it takes the subjunctive. If it said como usted pide (indicative), it would state a habitual fact: 'the way you usually ask'.

Why does the sentence use pida (from pedir) instead of a form of preguntar for 'ask'?

Preguntar means 'to ask' only in the sense of asking a question to get information. Pedir means 'to ask for' something or 'to request' an action. Since you are requesting that the nurse do a specific action, pedir is the correct verb here.

Is the word camilla (stretcher) related to cama (bed)?

Yes! The suffix -illa makes it a diminutive, so camilla literally translates to 'little bed'. In Spanish, this is the standard word for a hospital stretcher, gurney, or massage table.

Would it sound unnatural to put usted after the verb, as in como pida usted?

Not at all. In Spanish, it is very common and natural to place the subject pronoun after the verb in dependent clauses like this. Como usted pida and como pida usted are completely interchangeable in this context.