Web API without access to the bank makes sense?

Database and Web API are orthogonal concepts, so it makes sense in all cases that does not need a database.

It is rare, but it is possible, either because the data is already in the code, or it only does calculations and processing. Of course, if you do so it is unlikely that it should be Web, except for commercial issues. Actually I consider the use of Web API to be an abuse in most cases and this happened because most programmers do not know how to program and then buy something ready and the market decided that they could make more money by "renting" the API and avoiding piracy. People do not even think about it. No wonder the top 5 or 6 technology companies have exploded their market values in recent years because they created this new market and convinced people that it was cheaper, which is false in most cases.

Actually people do not understand what API is, and this has already been answered .

In general, doubt arises because people do not learn to program. They only decorate cake recipes, one day they ask, can I not for condensed milk. And this is the difference between the cook and the chef. The first one thinks that he is doing something interesting, but it is trivial and in the end it is only a manpower that one day can be replaced, the second one really does important things and it is imperative. Needless to say which one wins well. It is necessary to learn the fundamentals, to understand the reason for things.

A Web API allows you to make any services available to an end user connected to the Internet.

I do not agree to 100% with @Maniero, in the case of an online payment API for example, it will be complicated (impossible) for the lambda programmer to create the system from A to Z.

An online payment system (gateway)! This will require a database for the management of the different types of API cards and users and of course this database will not be accessible to the end user of the API and will be treated internally on the server.

An API that optimizes the size of an image it only needs the image and a treatment algorithm. There you will not need a database (just in case you have a paid plan (there you will have to save the customer data and use it, ...).

In all cases of professional API you have a database even if only to remember the access and treatment number, now it is clear that you will never open access to this database for the end user!