To what extent is it lawful to dispose of an amount allocated to an external agent who has not completed his work, and to use it to repair a private car owned by the user from the institution?

It is impermissible to dispose of another's right without their permission. What you have done is unlawful (haram), and you must repent, restore the right to its owner, and seek their forgiveness.

As for his failure to perform any work: if it was due to his negligence, then the matter concerns the institution, not you, unless they authorized you to act on their behalf. If it was not due to negligence, then neither the company nor you have the right to withhold his due or deduct from it.

Regarding your waiver of a car: if the institution stipulated this to you, then you are obliged to fulfill it. Otherwise, the matter is up to you. The important thing is to hand over the amount allocated to him, because your appropriation of it for yourself is a breach of trust (amanah), and Allah has commanded the fulfillment of trusts to their rightful owners.