I have built a team with main set silver sable from the DP set.
She is a wild card, and has the outlaws keyword.
I've made a soldier team.

Her trait is

WILD PACK, ASSEMBLE!: Adjacent friendly characters that share a keyword with Silver Sable can use the team ability she is copying.

Can she copy her at a outlaws, and then give it to adjacent characters?

Your always friendly to yourself.

If she can, she significantly better, not that she's already isn't.

A character cannot have the same team ability more than once. This extends to ATAs.

Now with Silver Sable, why would she need to Copy the ATA? If you are running a Theme Team you can simply assign the ATA to her. Then since her Wild Pack friends are Wildcards they could copy said ATA as long as it can be copied.

Scratch that,
Outlaws is uncopable,
But I could do marvel knights.
Could she share that?

The original post says his theme team is soldier which means he most likely has little to no wildcards other then Silver Sable on the team which is why the question is being asked.

As far as the question itself I do have to agree that Silver Sable can't copy her own ATA for her trait.

As long as the ATA does not say it cannot be copied then you can copy it.

Also the Outlaws ATA can be copied. Unless you were trying to use the DC Red Hood and the Outlaws ATA

She can also provide the Heroes for Hire ATA (wouldn't do anything, but can be copied)

The OP is not asking about copying the ATA's, He wants to use this trait: So Silver Sable has to copy a Team ability first. And as you pointed out she can not Copy her own TA.

I would say yes because according to page 20 of the rulebook, ATAs are possessed by the character and may be copied by wild cards.
And on page 19 of the rulebook, a wild card can copy TAs possessed by any friendly (which would include oneself)