EDIT:
Looks like LinkedIn has significantly changed their API (see https://developer.linkedin.com/support/developer-program-transition).
Therefore the accepted answer no longer works.
A bit of light googling suggests that a variety of "freemium" providers exist, such as https://clearbit.com/ and https://www.fullcontact.com/developer/company-api/, but YMMV.
ORIGINAL ANSWER:
LinkedIn has an API that meets your stated requirements.
The Company Profile API: Retrieves and displays one or more company profiles based on the company ID or universal name. Returns basic company profile data, such as name, website, and industry. Returns handles to additional company content, such as RSS stream and Twitter feed.
https://developer-programs.linkedin.com/documents/company-lookup-api-and-fields