"On the lam" doesn't seem to be anywhere in modern conversation except in reference to vintage gangster movies where everybody spits it out like rounds from a Tommy gun. From the context you know it means someone has become a fugitive from the law and is in hiding, but from where did it originate?
|Wanted gangster moll, Sugar Puss, holds up her tired tootsies after going on the lam. Ball of Fire (1941)|
"Lam" today seems to survive mostly in the term "lambast," meaning to beat with a cane or to argue angrily.
Just a note.