Can the Infusionsoft API be used in mobile apps, for example to place orders or retrieve a list of bought products for a customer?
The api is an underlying structure and can be used with any platform given the right preparation. The only consideration is the authentication process for OAuth that some prefer to avoid using with mobile apps. However, that not withstanding, your mobile app will function as the mobile platform you build it on would. The api would simply handle communications in the background and your app would be responsible for deciding what to do with the information on it’s own.
Infusionsoft Mobile is using the same API that you will be to interact with Infusionsoft. Here are some links so you can see what it will do what you are looking for (save some money by not having to build something that exists).