There is not an upgrade for the sparrow which allows it to fire a weapon. However, that does not mean you cannot kill anything with it.
You can kill mobs and teammates with the sparrow. To do this, you need to be at max speed, and need to directly hit them. The sparrow itself does not do any damage at this point. When you strike the teammate or mob they are hurled at speed. If they hit something they take the same type of damage as fall damage. This is how you can kill them, I have done it to both mobs and teammates (whoops!).
tldr; Causing a teammate or mob to hit an object by hitting them into it with your sparrow can cause damage or kill them. There is no firing option.