Let me get the approval of...

Let me get the approval of...
 

Let me get the Director to approve your application.

Déjenme consultarlo con el director para obtener la aprobación

Déjenme solicitar el visto bueno del director

Déjenme preguntarle al director si la apruebe o no

Other suggestions?

Even though it doesn't mention 'your application', I like the middle one.

The first one alludes to an agreement, or something else masculine, instead of an application (solicitud).

In the third, I believe you should use pedir instead of 'preguntar', to stay closer to your original version, followed by a subjunctive clause.

I agree with Rusty about the third one, although you can leave it as it is, except that "aprobar" must be in present indicative. ;)

Indicative rather than subjunctive? I thought the statement implied uncertainty as to whether the Director would approve it or not.

The 'si' clause doesn't take the subjunctive. You can change it to a subjunctive clause.