Will lead Jorah Heartsbane

Why does Sam get away with stealing this valuable artifact?

On the way to the Oldtown Citadel, Samwell Tarly stops at his family's fortress. Sam's relationship with his father is not good and besides, Sam brought a wild girl with him.

Sam's father Randyll Tarly hates Sam because he couldn't make a "good heir" out of him.

When Sam turned fifteen, his father bluntly told him that he was not worthy of the ancestral Valyrian steel sword Heartsbane of their house and that he would have to join the Night's Watch and give up his family name so that it would not stand in the way of Dickon's legacy.

Source: Samwell Tarly: A Wiki of Ice and Fire

During this visit, Sam's father kept teasing him about going to the citadel "about the big tasks another Men to read ".

However, when Sam leaves the family mansion, he takes the ancestral Valyrian steel sword of their house with him.

It should be clear that Sam, Gilly, "their" baby and such a precious artifact as an ancestral Valyrian steel sword are disappearing at the same time (or almost).

Why did Randyll Tarly never investigate the sword's disappearance and / or track Sam to regain it?

Möoz

He it does simple, nobody noticed or mentioned it

It's hard to tell whether or not Randyll (or another member of the household) noticed it, as it is never mentioned again afterwards.

The fact
is that Sam took the sword and probably still has it in Winterfell. That's all we know for sure.

The explanations
are they either:

  • Randyll and everyone else didn't notice
    • Which is completely atypical for a Valyrian Steel sword that disappears without notice, and also inconceivable for Randyll (one of the most pragmatic and proudest men in Westeros) not to notice.
  • Randyll has it noticed but not mentioned on the screen
    • This can be understood as Randyll doesn't have a lot of screen time at first. Also, I would doubt Randyll would want anyone to know about the missing sword as it matters.
    • Want you Tell people like Jaime and Cersei Lannister that your denounced and embarrassed son and his "wild" girlfriend stole the ancestral sword (House Tarly's most important possession)?

From here
There will likely be two results:

  1. The showrunners forget to mention or leave out this detail as it is not worth spending screen time on
  2. Sam can explain it to Jon as soon as Jon returns to Winterfell and Sam conveniently gives him a fantastic White Walker Killing weapon

Anyway, let's wait and see.

Aegon

+1. A suggestion, maybe he was just pleasantly surprised that his cowardly son showed courage once and he was pleased about it. Samwell is a Tarly, as long as he has the sword it is still in the family. Would have been an "emotional scene" if they'd shown Randyll Tarly stealing the sword and wrinkling his lips in the same smile he gave Dickon when he decided to burn with his father

Flater

Another suggestion: Randyll already regards Sam as irreparably incapable of being a soldier or living a meaningful life, and this shows in how he assumes Sam is too Nothing is incapable (even of various things, not just as a soldier). It's possible that Randyll suspects Sam stole the sword but didn't keep it ("As if that pathetic weasel had some use for it. Laughing!"). Maybe he sold it, maybe someone has already taken it from "that weakling". So there's little point in chasing Sam (who won't get Heartsbane back anyway), and he'd rather never see or hear from him again.