'External training unit' sounds fine to me, although it doesn't necessarily imply that contract negotiations take place. The name merely implies that there's a group of trainers who are sent to various off-site locations to do training (and it's probably assumed that there's a contract with the outside company to do so).
Are you trying to translate the name of this department from another language? If so, I would wonder what the literal translation from that language into English would be.
(not a teacher, just a language lover)
- For Teachers