Does the free version allow companies to use it?

Our department needs a local repository to store ASCII type files (not code, for version control and data sharing only, no other functionality is required) and the service will be deployed on a local server.

I see that the GitLab source page says “Completely free and open source (MIT Expat license)”, but I also see that the official page says “Pricing”.

Does this “Pricing” refer to the price of the online GitLab service or the price of the GitLab software?

It refers to subscriptions for use of eg: gitlab.com or your own gitlab server. If you don’t need the functionality, then you can use gitlab for free without the additional functionality that premium or higher subscriptions offer.