I think it depends.
Asking for (Web) APIs that …
- (1) … do a specific job: on-topic.
- (2) … provide specific data: off-topic.
(1) Asking for an API that "corrects spelling errors in a given text" would be on-topic. That would be similar to asking for a spell checking Web app recommendation.
(2) Asking for an API that "gives the current weather in a given city" should be off-topic. That would be similar to asking for a weather website recommendation.
{{ When a specific service (for example, a search engine) provides a "meta" API that allows others to provide APIs on top of that, asking for an API for that specific service should be on-topic, as it would be the case (1): While it’s, of course, providing data, the question is about accessing this specific data set. But this is probably a rare case, as there is usually only one API for a specific service (and in that case, it wouldn’t make sense to ask for it). }}
The difference between Web apps and APIs is the access method: browser+HTML vs. API client+[XML, JSON, …], so I think the same rules should apply. If a Web app question is on-topic, the same question asking for a Web API instead would be on-topic, too. When a question would be closed because it’s asking for a website recommendation, the same question should be closed if it would ask for a Web API instead.