Combat damage does not use the stack. So you cannot use the unmodified creatures to get both triggers.
However, if you give Etrata, the Silencer First Strike, you will be able to get both triggers in the same combat step. Here's how:
First Strike works by creating an additional combat damage step. The combat damage step, in simple form, starts with damage being applied, then gives players priority (i.e. they can activate abilities or cast instant-speed spells).
So, here's how you'd time this:
Etrata, the Silencer is unblocked and deals damage during the First Strike Combat Damage Step. This is also going to trigger her ability, which will end up on the stack first.
Once First Strike combat damage has been dealt, you now gain priority since you are the Active Player. The stack has Etrata, the Silencer's ability on it.
Etrata, the Silencer is still "attacking" and still unblocked--even though she's already dealt damage, until the end of combat step she's still considered an attacking creature.
While Etrata, the Silencer's ability is still on the stack, you activate the Ninjutsu ability. As part of the cost for the Ninjutsu ability, you return Etrata, the Silencer to your hand. The stack now has Etrata, the Silencer's ability and Yuriko, the Tiger's Shadow's Ninjutsu ability on the stack.
Etrata, the Silencer's ability resolves and you get to exile a creature with a hit counter on it. This still works even though Etrata, the Silencer is no longer on the battlefield to be shuffled into your library--she is safe in your hand, ready to be re-cast.
The First Strike Combat Damage Step ends, and you move to the regular combat damage step.
Yuriko, the Tiger's Shadow will do combat damage during this step, which will trigger her ability.
This is pretty important--cards like Akroma's Memorial will not work with this plan--they also give Yuriko, the Tiger's Shadow First Strike, so she would not be able to do damage in the Regular Combat Damage Step. If you gave all your creatures Double Strike, this plan would still work.
I can't think of any Blue/Black cards that would really work well for this plan--there's some equipment, but you'd have to re-attach it to Etrata, the Silencer every time you wanted to perform this plan.
Ninjutsu is not casting the card--it's an ability that puts the card directly onto the battlefield without using the stack. Like Derevi, Empyrial Tactician's ability, the Ninjutsu ability bypasses the Commander Tax.
October 11, 2018 4:16 p.m.
To provide a slight bit of Etrata errata to my previous post:
"However, if you give Etrata, the Silencer First Strike, you will be able to get both triggers in the same combat step" should properly say “same combat phase.”