The primary risk is not for the provider of the API, not the consumer. There may be some risk for the consumer on the copyright side but not necessarily, but there may also be a contractual risk (see below).

In US law, there are fair use exceptions that may allow use of a copyright protected logo (or any other copyright protected material) without causing infringement - but those are relatively narrow. It may also be that a brand whose logo is generally available is OK for that logo to be used, without it taking action for infringement - but that doesn't necessarily remove their right to ask for uses that they are not happy with to be stopped.

So simply using a logo may not be copyright infringement - and, for example, the use cases presented by the service here - company logos in transaction statements, etc - may be allowed by a brand, but if they were unhappy with how someone was using their logo they would be within their rights, under copyright, to ask for that use to be stopped, or ask the person or organisation using the logo to enter into a licensing agreement.

As soon as you start charging money for someone else's copyright protected works without a licence that allows you to do that, it is almost certainly copyright infringement. So by making a copyright protected logo available via a paid API without the copyright owners permission, you're very likely immediately opening yourself up to being sued, and in the US you could probably be sued for statutory damages for each time you've infringed. Which would mean that if a logo has been served through a paid API to 10 different customers, suddenly you could have a potential liability of $1.5 million.

So there may be a issue for the consumer of the API depending on what they are doing with the logo, but it's unlikely that they would be pursued for damages. But it is 100% an issue for an organisation providing an API that explicitly makes money off someone else'e copyright. And that means that if the terms and condition of the API say "we have the rights from the copyright owners to provide the logo to you" then they are also in breach of contract with the API consumer. If they don't include wording to that effect in their terms of use, then use of the service would not pass due diligence by anyone serious. So if you're a financial services application and you want to display logos in your application, you'd want to be certain that your use of those logos is legally permitted, and you'd look for that assurance from the API provider. If you don't, then there's a potential liability, which is what I meant about a contractual risk. If you, as the application provider, are making an onward warranty that your application is fully licensed (for example, as a white label version to an enterprise customer) then that could present a risk for you as the application service provider, because of the API's non-compliance. But that's an adjacent thing, really.

Copyright gives a couple of fundamental rights: first of all to allow you to control how your work is copied, and by whom; second, to allow you to choose who you allow to copy your work, and on what terms. The second means that you can grant someone a licence you copy your work in exchange for money.

If someone else starts copying your work and charging money for it, they are very clearly in breach of the protections copyright gives.

So in this case the service is explicitly creating copies of something protected by copyright for money. There's presumably a copy being made at the point it is scraped - which could be covered by fair use - but certainly at the point that the logo is being served to a paying API consumer a copy is being created and the API being charged for that copy, and that would almost certainly negate any fair use argument.

Unfortunately, the only real way to offer a service like this as a paid API is to go and get agreement from every brand whose logo you want to use. Now, it may well be that they would be happy to do that, if you can also offer them a brand protection element to the service - e.g. something like "the only way you can use our logo online is via this API, either as a dynamically served logo (which can be revoked at any time) or as a watermarked logo, tied back to the API consumer (who assumes liability for infringement)". Add in search and enforcement for infringing uses of the logo (and other brand assets) and brands may happily licence the logo to the API provider for free, as part of a wider service.

There's a whole other - potentially bigger - issue to do with trademarks. That's harder for brands, because if you don't enforce infringement of your trademark then you may lose the rights in the trademark. However, that'a a much more complex area and varies a lot from territory to territory.