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

1. The water starts at the top in vessel 9.
2. 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.
3. The water flows to vessel 7, where it has the option to go left to vessel 6 or right to bucket 5.
4. The outlet from vessel 6 to bucket 4 is blocked, so the water cannot flow there.
5. The water flows directly from vessel 7 into bucket 5.

Therefore, the order in which the vessels fill up is as follows:

1. Bucket 5
2. Bucket 3