Dear Colleague,
If you engage any agency or service provider to service you exclusively, then YES. Whether you pay them kilometer-based or otherwise does not matter to the Labour Laws compliance.
Service Provider as Contractor
The Service Provider will be treated as a CONTRACTOR in the eyes of the law, and all compliance has to be done by the Service Provider management, and you become the Principal Employer in such an arrangement. As the agency has 25 plus employees or more probably, then they become an "Establishment" in the purview of the law, and they have to be compliant, and you have to ensure it to avoid statutory non-compliance risks.
Engaging Vehicles by Managers
Some companies allow the Managers to engage their own Vehicles, and they have to enter into a contract with different service providers. Here the company pays Petrol/Driver Salary and Vehicle hire charges to their Managers upon producing the valid Vehicle lease agreement. This will, to some extent, avoid Labour Laws compliance by the Company.
Compliance Responsibility
But in the given situation of your arrangement, compliance has to be done by the Service Provider, and the Principal Employer has to ensure and monitor. You need to demand their PF Code/ESI Code/Contract Labour License and so on and ensure monthly remittance by asking for proof of copies of compliance documents online.