Yes, tackle can mean that ("to 'take on'"). But that doesn't seem to be the meaning in your suggested sentence. It seems to be to "seize and throw down" Al Qaeda at its bases. But I might be reading something into that sentence that's not there.
The meaning is given in B3 in the forum dictionary. It has nothing to do with sport, although it may come from a sporting metaphor (e.g. rugby in the UK).
We tackle/take on/try to solve problems or difficult tasks. Sometimes "deal with" fits.
If you have an enemy and engage with them, you take them on. This means more ore less to fight them or struggle against them.