There are many ways to approach this problem. You may find it easiest to work through it backwards.
In this diagram, the starting point is at S, and a trip around Flargg passes through points A, B, C, D, E, F, G, H, and I. (These points mark intervals of 1000 kilometers along the equator.)
Recall that a Flargmobile can travel 2000 kilometers on a tank of water, and that it can carry two tanks at a time in addition to its passengers. Thus the maximum distance it can travel without refueling is 4000 kilometers. If there were two spare tanks at point C (3000 kilometers from the start at S), another one at G, and a fourth one at I, as shown in the diagram, it would be possible to circumnavigate Flargg by starting with two full tanks from S and picking up these four spare tanks along the way. Thus the puzzle is reduced to that of delivering spare tanks to these three points. (This is only one possible solution; there are others.)