Some good suggestions so far.
A few of my thoughts....
Check the the passenger door jamb flex harness for breaks. You can open the door and when the puddle light should be on, wiggle the rubber boot where the door harness goes from the inside body to the door. If wiggling it momentarily makes it come on, then you have a break in a wire in that harness.
Does it look like the mirror or door has been apart due to an accident?
If you remove the small sail panel trim cover on the inside of the door opposite where the passenger outside mirror mounts to the door, you'll find an electrical connector C615. I didn't see what year SHO you had, but based on your profile picture I looked up the wiring schematics for a 2015 Taurus SHO.
The BCM feeds power to both mirror puddle lamps through fuse F12, a 15 amp fuse and then splices at Splice 384 then through inline connector C3139 under the passenger side of the dash and onto the mirror connector C615.
When the puddle lamp should be on, you can check with a volt meter and/or test light for 12 v at pin 5, on the VT / Violet colored wire and ground at pin 7, on the BK/YE Black/Yellow wire.
If it lights up with power and ground at the connector, then you have an issue with the internal mirror harness or socket for the puddle lamp bulb. If it doesn't light at connector C615, then you need determine which is missing for the bulb, power, ground or even possibly both. Then you can back track from there. The flex harness is something that can be problematic on door related wiring issues, particularly the driver's door, as it gets the most use. Obviously, the passenger door is where I would focus.
Hope this helps.
Mike