jak to będzie

wysłałam to
jak tylko mi przyślą to ci odeślę

>wysłałam to

I have sent it.

>jak tylko mi przyślą to ci odeślę

I will send it to you as soon as I get it.

Once they mail me I will send it to you.

"mail" is a transitive verb. It requires a direct object in order to make its meaning clear and complete. In your sentence the direct object is "me" which would mean that "jak tylko wysla mnie" :)) gdzie cie beda wysylac?
You need to put "it" there and change it to "once they mail it to me I will send it to you". Ale to dalej nie ma sensu, bo to ze oni to wysla (mail) nie oznacza jeszcze, ze ty to dostaniesz (moze np. zginac)

Once I get it, I will send it to you.

how do you know it's an email?

I don't know. And I didn't assume it was an e-mail. I wrote my post in response to a previous post by krolikk. He/she used the verb "mail" as "przyslac" in Polish. There is nothing about e-mail there.