The short version
Reciprocal play is a courtesy between clubs. You do not request it yourself; your head professional contacts the other club's professional on your behalf.
Some clubs belong to formal networks with set terms. Many arrange it case by case.
How to ask
Give your professional several possible dates and the size of your group. Expect to pay a guest fee and caddie fees at the host club.
Requests for peak weekends in season are rarely granted. Weekday afternoons in the shoulder season are the best window.
On the day
Arrive early, check in with the pro shop, follow the host club's dress code and phone policy, and send a thank-you note to the professional who arranged it.