Is it permissible to shave during sheloshim (after shiva) for the sake of ones job?
There is room to permit this, if shaving is required for maintaining one’s job or one’s income.
The custom of refraining from haircuts during the sheloshim is generally applied even to shaving, but there is some room to distinguish between them, because shaving (which is done daily) does not involve the festive nature of a haircut.
In addition, it is possible that reasons of losing one’s parnasah will be sufficient to waive the custom, as we find in a number of poskim concerning Sefiras Ha-Omer (see Kaf Ha-Chaim 493:19; Iggros Moshe Orach Chaim 4:102).
I have seen cited from Rav Moshe Soloveitchik that he permitted somebody to shave within the sheloshim for parnassah reasons, though his reasons for this were not given.