Historically interesting note: Nazi Germany kidnapped thousands and thousands of Polish children and handed them to German parents to be raised as Germans. Their decedents usually don't even know about this.
Back to the topic at hand: you have to distinguish between material and formal cooperation with evil. I don't know the Kagi situation (if they're just indexing images for Yandex to improve search results, then I don't see how you have a real case here; even calling this remote cooperation with evil -- something that is generally impossible to avoid -- seems like a stretch). But let's say a company is doing something like making financial contributions to some organization doing something immoral. While you can boycott a company for that reason, you are not generally morally obligated to do so. And in practice, it usually has no effect. It's also unjust to saddle people with a burden of guilt they do not actually have. This is called rigorism.
Back to the topic at hand: you have to distinguish between material and formal cooperation with evil. I don't know the Kagi situation (if they're just indexing images for Yandex to improve search results, then I don't see how you have a real case here; even calling this remote cooperation with evil -- something that is generally impossible to avoid -- seems like a stretch). But let's say a company is doing something like making financial contributions to some organization doing something immoral. While you can boycott a company for that reason, you are not generally morally obligated to do so. And in practice, it usually has no effect. It's also unjust to saddle people with a burden of guilt they do not actually have. This is called rigorism.