Is it permissible to use mediation to secure a teaching position for one's wife in the Ministry of Education, given that appointments are made through mediation and external interventions, and would that be considered a sin?

It is permissible to obtain a government job through 'wasta' (mediation/connections) for someone who is qualified to perform it and if it does not result in injustice to anyone.

If the use of 'wasta' is to remove an injustice from your wife or to secure her rightful due, then there is no harm in that.

However, if 'wasta' leads to your wife bypassing those who are more deserving than her, then this is an injustice and is not permissible.