Can a soldier call Staff Duty to sign in or out for leave?

That would be a local policy issue. At my post only E7+ can call.

That is in the hands of your Command- their SOP will state which you can do- most require you to sign in person.

It is different at each post, but the general rule of thumb is SFC and above can call in. Honest truth, I have never been in a unit that allowed anyone below that rank to call in. Check your policy letters, Leave guidelines are always outlined in a policy letter.