Interestingly, it does seem to be the case that a Vampire cannot charm a Blind target, or any creature that cannot see the Vampire
You have correctly quoted the Vampire Charm ability as stating:
The vampire targets one humanoid it can see within 30 feet of it. If the target can see the vampire, the target must succeed on a DC 17 Wisdom saving throw against this magic or be charmed by the vampire [...]
Compare this to the charm person spell which has no such requirement:
You attempt to charm a humanoid you can see within range. It must make a Wisdom saving throw, and does so with advantage if you or your companions are fighting it [...]
And now compare it to the Eyes of Charming which do have this same restriction:
These crystal lenses fit over the eyes. They have 3 charges. While wearing them, you can expend 1 charge as an action to cast the charm person spell (save DC 13) on a humanoid within 30 feet of you, provided that you and the target can see each other [...]
This shows that there do at least exist charming effects that also require two-way line-of-sight.
It should be noted that being under the blinded condition does indeed make the creature unable to see:
A blinded creature can't see and automatically fails any ability check that requires sight.