Can my sister-in-law become executive over my dead husband's estate even though he did not give her that right and he has 3 grown sons and a wife?

If he died intestate then the she would probably be third in line to be administratrix.  The wife would have first right and then the kids to serve as administrator/trix. 

I am so sorry for your loss.  I am assuming that your husband died intestate or with out a Will when you say here that he did not name her as executor.  Then she is applying to the court to be appointed.  She can and you can object.  You can ask the court to be appointed as the personal representative of your hsuabnd's estate and your children could as well.  In some states the spouse is the only one who can apply.  Please seek legal help.  Good luck.