To determine the order in which the vessels will fill up, we need to examine the paths the water will take:

The water starts at the top in vessel 9.
From vessel 9, the water can go left to vessel 8 or right to vessel 7. However, the outlet from vessel 8 to bucket 1 is blocked, so the water cannot proceed in that direction.
The water flows to vessel 7, where it has the option to go left to vessel 6 or right to bucket 5.
The outlet from vessel 6 to bucket 4 is blocked, so the water cannot flow there.
The water flows directly from vessel 7 into bucket 5.
Therefore, the order in which the vessels fill up is as follows:

Bucket 5
Bucket 3