What is the ruling on keeping a client's money that was paid to the institution to settle a debt owed to the institution, knowing that the institution is indebted to the employee and procrastinates in paying his debt?

It is permissible to take the agreed-upon amount of your right from the institution without their knowledge if they renege on the agreement, provided that you do not exceed this amount. This is known as the issue of "Al-Zafar" (taking one's right by force).