I'm not entirely sure but I believe it's the classic DuckTales cartoon from the late 80s. Which episode is it and does it hold a meaning for the movie?
All the clips are actually from the intro for season 1 of the original DuckTales. The clips that follow prove this point.
The first time we see them watching it is in Ohio on the TV in the background.
The episode here is season 1 episode 6, "Send in the Clones". This episode has clear parallels to Black Widow in that the family are a sleeper cell at the start of the film. On top of that, as revealed later in the film, there are Widows as sleepers all over the world. To an extent the Widows themselves, stripped of free will, are clones of each other as well.
Everyone begins seeing double when Magica DeSpell gets the Beagle Boys to help her steal the Number One Dime, disguising them as replace the nephews. Can the nephews expose them before the dime is stolen (and before Mrs. Beakley loses her mind)?
As part of this sequence we see the intro clip over to the next scene. This appears to be an intro only clip.
The second time we see them watching it is when we see Nat being forced to watch it. Presumably to help if she needs to go undercover again.
Again as far as I can tell this particular clip is from the intro only.