Smartphone app query

It depends although the answer is probably going to be yes you do need a website as well.

If the users install an app with a menu and prices then they have to get the prices to you to include in the app. In theory they could email you the menu and prices etc. but I would assume that at some point they would want to update the menu / prices without installing a new version of the app.

Option 1, the cleanest way would likely be to have a regular website where the prices and menus can be updated, and then have the phone app query the website to get the updates periodically when online and cache them to keep it up to date.

Option 2 you could have the restaurant email you the updates and you could package a new version of the app and publish the new version and then the user would get a notification to update the app and when they install the new version they get the new prices/menu. Technically no website would be needed for this.