Shouldn't +vigor from items add to blessed hammer damage?

The above is mostly correct. Mostly in that there is one exception:
Druid summons. They actually get the full synergy bonus from one another based on total skill level, not just hard points.

For everything else, synergy bonus is based on the number of actual hard skill points invested and hence caps at 20 and does not improve with gear.

Correct. It bumps conc and in doing so bumps hammer damage while conc is active.

Apart from that, you only get more hammer damage either from getting more skill points into hammer itself or by investing more (up to 20) hard points into the synergies if you haven't already.

Side note:
Technically speaking, there is one more exception to the hard point rule for assassins as well. Not for the sin herself but her shadows follow the same concept of their skills synergizing with their other skills after said other skills have been used. They actually base that synergy on their own final skill level being used which allows a shadow to be 'trained' to potentially even higher trap damage for example than the assassin herself. It doesn't change anything on the assassin though, only works for the shadow itself.