From a VAT perspective, I don't think that the use of the vehicle is particularly relevant to the VAT treatment of the installation. The vehicle belongs to the employee, so the only VAT that the employer is entitled to reclaim in respect of business use is the fuel element of any mileage allowance paid. And even that was challenged by Europe some years ago, on the basis that the fuel supplier's supply was more likely to be made to the employee. This led to the current requirement for employees to retain fuel receipts.
However, back to the question! If the employer contracts for the work to be carried out, and then makes a charge to the employee, the employer becomes a step in the supply chain. The installer supplies services to the employer, and the employer supplies services to the employee. The employer can reclaim input tax, because they are making that onward taxable supply. Theoretically, the charge made to the employee could be less than the cost to the employer. Although if the charge to the employee was too negligible, HMRC might not consider it to be a business activity, which would then prevent the input tax recovery.
Without a charge to the employee, it's hard to see what business activity the installation could be attributable to, so input tax recovery becomes unlikely.
As for the "rewarding the employee" angle - perhaps this is derived from the employee entertainment rules in section 3 of notice 700/65? If so, that relates specifically to "events", so doesn't really apply here.
I realise this is a VAT forum - but remember that if the employer bears some or all of the cost of these works, there may well be a taxable benefit to the employee, which would be reportable on the P11D.