Is there a specific word for "causing someone to faint"

Solution 1:

Stupefy seems to fit your requirements. It is a single transitive verb that comes closest to knock someone out.

1. to put into a state of little or no sensibility; benumb the faculties of; put into a stupor.
2. to stun, as with a narcotic, a shock, or a strong emotion.
3. to overwhelm with amazement; astound; astonish.

as in

He stupefied her with his next statement. She sank to the floor in shock when he told her he was her father

Note to downvoter

Although having the same root, stupor is not stupid:

stupor (Oxford dictionaries)

noun a state of near-unconsciousness or insensibility. "a drunken stupor"

synonyms: daze, state of stupefaction, state of senselessness, state of unconsciousness; inertia, torpor, insensibility, numbness, blankness, oblivion, coma, blackout; Scottish dwam

Even Harry Potter agrees -

Stupefy (from Wikipedia, list of spells)

(Stunning Spell, Stupefying Charm, Stunner) Pronunciation: /ˈstjuːpɨfaɪ/ stew-pi-fy

Description: Puts the victim in an unconscious state. Manifests as a jet of red light.

Seen/mentioned: First seen in Goblet of Fire, used by Ministry officials at the Quidditch World Cup and later against Barty Crouch Jr.. Also seen used by a number of Ministry officials against McGonagall in Order of the Phoenix. It is also taught by Harry in his Dumbledore's Army meetings and used extensively during the Battle of the Department of Mysteries against the Death Eaters. Is seen by some, including Harry himself, as the basic spell for fighting. Death Eaters, Ministry Officials, Order members and students all seem to refer to this spell as their preferred attack.

Solution 2:

You might describe someone as having a "syncopal power" over someone else. I don't think there is a word which describes the effector of syncope but I'm sure a portmanteau could be coined if you really have an application for such a word.

Solution 3:

As a single word, daze can be used in this sense.

Example:

Percy Phelps sank back in his chair. 'My head whirls,' said he; 'your words have dazed me.'

It is not a single word but you can consider make someone swoon.

Example:

Then one morning in late August, 1947, I got a phone call from Bob. My blood pressure must have gone up 100 degrees. Just hearing his voice again made me swoon.

Faint is (or was) a transitive verb also but the usage is obsolete:

To cause to faint or become dispirited; to depress; to weaken.

« It faints me to think what follows.» Shak.

Solution 4:

• to knock out
• to stupefy
• to stun
• to cause to black out