I can find in the API docs how we can get the products to show them on our website, and I also found that I can create a manual payment in case users don't want to pay with a creditcard.
The docs say you need to set manual payments as "paid" as soon as you have gotten the money.
Can this also be done with an API? if so, with which one?
To mark a manual payment as paid through the API, you can use the Orders Transaction API to "capture" the payment. Here's an example of what that call will look like:
You can find more details about in our API docs here.