He is in the Nights Watch. Remember? He was sent to the Citadel by the Lord Commander to become a Maester?
That's one reason for why he can't become a Lord (or hasn't become a lord yet) but that might not even be the case, the time is not at all right, he had also vowed to never be with a woman, but he broke that vow, I do not see why it will not be possible for him to break away totally from the Night's Watch vow after the Great War ends (if it does and the living wins.)
Most of the answers above talks about his vows, totally understandable but until now Sam didn't even know his father and brother are dead, neither did Daenerys have any idea that Lord Randyll Tarly had one more son who could be the Lord. What about the Night's Watch vows? Oh, the Wall has fallen, the ancient order of the Night's Watch may not even be relevant now so how does the vows will matter after the War ends? If Jon Snow after becoming the King wants he can name him the Lord to at least save one great house from extinction, just like I believe he will legitimize Gendry to save the House Baratheon.