Then would they launch an invasion? Or would they try to support the rebel force?
There are examples where a country that has interest in another country's piece of land would invade the moment chaos reigned in the latter. Of course, this is so that the said chaotic country won't be able to mount an effective resistance in the event the first country invaded.
A perfect example was the Polish Intervention during the Russian Times of Troubles in the late 16th - early 17th century. With Ivan the Terrible's successor Fyodor having had no legitimate heir and Boris Godunov succeeding him, various boyars vied for the Tsar's throne after Godunov's death. With chaos reigning and the appearance of the first False Dmitry, Poland decided to intervene and invade Muscovite land.
Another, more modern, example would be the Turkish invasion of Northern Syria during the height of the Syrian Civil War. With the government of Assad busy dealing with rebels and ISIS, the Turkish army was free to invade their neighbor under the pretext of stopping and punishing 'terrorist' Kurds.
Of course, to sow discord and chaos in one place, one has to take note of the societal make up of the region. So for this, one has to know who and what to support. The Allied Powers of WW2 demonstrated this in Yugoslavia, where they sought to defeat Germany in the Balkan country by choosing who to support. In the end, they supported Tito's Partisans over the Chetniks.
However, I was wondering if the rebel lacked manpower to the point that taking the government was impossible and the people were too weak or don’t want to stand in front of machine guns, then how do the corrupted government be taken down?
Any rebel force should take note that heading into conflict unprepared guarantees defeat. So, they should prepare for war as covert as possible. The preparation may take long, but as long as the movement stays under the radar, the fire of revolution remains alive.
But, let's say for argument, how can an unprepared rebel force and a subservient populace take down a government? One good example to look at would be the Philippine Katipuneros (a secret revolutionary movement during the late 19th century Philippines aiming to liberate the colony from Spain). The movement was discovered prematurely, so the founder, Andres Bonifacio, have to launch the uprising unprepared.
Results are mixed. Some provinces rebelled successfully, but the overall phase 1 of the Revolutionary War nearly failed, had it not for the intervention of the Americans...who would be the next colonial masters of the Filipinos in the aftermath of the Filipino-American War.
A government can fall, but they never disappear. There are just replaced by new forms. Governments are just power structures of administration and governance. We humans are social creatures by necessity, so they will always continue to exist. There will always be leaders.