Is using Social Ninja allowed under LinkedIn’s terms of service?
Yes, when used as designed. Social Ninja connects to LinkedIn through OAuth and the w_member_social scope, which LinkedIn’s Posts API documentation describes as “post, comment, and like posts on behalf of an authenticated member”. Posts are created via the official Posts API under your authenticated profile, at the time you scheduled, with content you already approved.
That puts us firmly inside the productivity-tool category and away from the browser automation, scraping, and bot tooling that LinkedIn’s User Agreement prohibits in §8.2.