Although this series was inspired by the first MCU Avengers film, it isn't set within the MCU. Instead, it's apparently set in the same universe as the Ultimate Spider-Man animated series, which was designated Earth-12041 in Secret Wars: Official Guide to the Marvel Multiverse #1.
Since it isn't set within the MCU, the Infinity Stones in this series can be somewhat different from the ones in the MCU. They're clearly not exactly the same, as these stones speak to people around them, attempting to seduce and corrupt them, which doesn't apply in the MCU.
Also, these Infinity Stones don't have the same colour designations as the ones in the MCU. In Avengers Assemble S02E13, we see Thanos using individual stones, and the Time Stone is shown to be orange in this universe, whereas it's green in the MCU. The Mind Stone is shown to be blue in this universe, whereas it's yellow in the MCU. And the Space Stone is shown to be purple in this universe, whereas it's blue in the MCU.
In the universe the mainline comics are set it -- Earth-616 -- the Infinity Stones (or Infinity Gems, as they're also called in the comics) aren't portrayed as being directly harmful to the wielder, the way they are in the MCU. For example, in She-Hulk Vol. 1 #12, She-Hulk was able to wield the Power Gem in her human form, Jennifer Walters, without feeling any pain or sustaining any injury.
She-Hulk Vol. 1 #12 (April, 2005)
It seems likely then that the Infinity Stones in the Avengers Assemble animated series take after the ones in the comics, in this respect, rather than the ones in the MCU. Natasha's S.H.I.E.L.D. training was given as the reason that she could resist the stones' corrupting influence, but it doesn't make sense to think that this would also allow her to resist being physically harmed by the stones.
WIDOW: Hulk, these stones. They corrupted Tony and they'll do the same to anyone who comes in contact with them.
HULK: Yeah, like you.
WIDOW: No. My S.H.I.E.L.D. training can keep the stones from taking over my mind, but not for long.
Avengers Assemble - S02E12 - "Widow's Run"