No, your customers do not have to download the Stamp Me App to take part in your loyalty program, if they do not want to.
You can manage their account from your web-based Merchant Console where you can create members, apply stamps, see the number of active and all time stamps a member has, and mark rewards as having been redeemed. Many merchants choose this way of operating. The Merchant Console is optimised to be used on mobile, ipad or desktop.
However, there are many benefits to encouraging the member to download Stamp Me:
So they can see their stamps and rewards at any time, without having to ask you
So you can send them communications such as SMS & Push Notifications
So they can take part in engagement features such as join the Birthday Club and play Scratch & Win games