Can a single company enroll in both the Apple Developer Program and the Apple Developer Enterprise Program? As per Apple Developer Program website, an organisation can enroll in Apple Developer Program and Apple Developer Enterprise Program.

Is it possible for a company to be enrolled in both at the same time? For instance, if the company has apps that are meant for internal distribution only, but also has other apps which are targeted at customers and are distributed via the App Store.
Yes, It is possible from a single company to be enrolled in both the Apple Developer (Organisation) and the Apple Developer Enterprise program.

Kindly note that you will require a separate Apple ID to enroll in Apple Developer Enterprise Program. Quoting from the Enrollment Support Page:


  Enrolling in Multiple Programs
  
  The Apple Developer Enterprise Program and the iOS Developer University Program each require a unique Apple ID than that used for any other membership. You’ll need to set up a new Apple ID and enroll separately. If you currently have an iTunes Connect account to sell another media type (books, music, TV content, or movies), you cannot use the Apple ID associated with that iTunes Connect account to enroll in any other programs. You’ll need to set up a new Apple ID for all other memberships.


Disclaimer: I have worked with a company as a developer which has enrolled in both the programs. I was not the legal entity responsible for managing the accounts. I recommended you go through the Membership Support documents before proceeding.