There are 9 Vessel Fragments. For every three that you collect your maximum Soul is increased by ⅓.
5 Vessel Fragments can be found in static locations in the world. Many require the Mantis Claw, Crystal Core, and/or Monarch Wings to reach.
2 can be bought from Sly in Dirtmouth.
1 is awarded by the Seer in Resting Grounds after collecting 700 Essence with the Dream Nail
1 is obtained by donating 3000 geo to the fountain in Ancient Basin. This map
is a great resource for finding everything in the world. Use the control panel on the right to uncheck everything except "upgrades" so you can see the location of all Vessel Fragments (as well as Mask Shards, Pale Ore, spells and permanent upgrades). You can also choose to use the sketch map image (actual in-game map) rather than the detailed map image.
Additional information for each Vessel Fragment can be found on the Hollow Knight wiki