Why can't the Avengers go back in time to 2016 and bring Black Widow from 2016?
I guess they can. What they can't do is, you know, getting her back from the dead because getting the soul stone requires eternal sacrifice. Hulk did try to bring her back from the dead but failed as he states that to Cap.
But the act, if done, would create another timeline.
I think the way it happens in MCU is that you can go back in time and acquire things and come back to the present. This way multiple original Infinity stones could be brought to present time..
Thanos, Gamora came to the future (from 2014) even though Thanos and Gamora are -- or I should say were -- dead in current timeline. So, yes, it is certainly possible.
As mentioned in comments by IronSean:
And that other timeline wouldn't have a Black Widow from that point on. And since she was instrumental in getting the Soul Stone to stop Thanos, it dooms that timeline