Would everyone be considered sinful if a commission were paid to non-Muslim employees in large companies in exchange for obtaining labor supply contracts, knowing that these employees only contract with those who pay, regardless of quality, and there is no competition based on quality among service providers?

The forbidden bribe is that by which one attains what is not rightfully theirs. However, if it is used to attain a right, or to avert harm or injustice, then it is permissible for the giver (Rashi) but forbidden for the receiver (Murtashi). If those who enter into contracts do not contract except with those who pay them, then there is no impediment to paying, and the sin is upon them, not upon the one who pays, if he finds no alternative.