Is it permissible for a soldier to take money from his work to cover his car expenses, or to take the difference in price between what he buys for work and the market price, knowing that he incurs financial losses due to his work?

The default principle is that the party should bear the cost of services rendered for its benefit. However, if the soldier agrees with his recruiting authority to bear the expenses of transportation and travel while purchasing items he is tasked with acquiring as their representative, then the agreement must be honored. This is based on the Almighty’s saying: {O you who have believed, fulfill [your] contracts}, and the Prophet’s (peace and blessings be upon him) saying: "Muslims are bound by their stipulations."

Therefore, it is not permissible to take money for this purpose except with the permission of the authorized party.

It is also not permissible to take the discount for oneself without the knowledge of the one who commissioned you to make the purchase.

The soldier can request from his superior compensation for any damage or losses he incurred.