I've never tried connecting an external storage device to my vita/tv, but, coincidentally, I did need to find a power supply replacement for a WD drive a few days ago.
https://support-en.wd.com/app/answers/detail/a_id/8216
That page lists the power supply requirements for their drives. USB is standard 5 volts, but the Vita TV may not be able to provide the needed amperage like Hayato213 suggested. In that case, you should try using a powered USB hub, which allows you to power the drive using an external power supply.
Here's an example:
https://www.newegg.com/p/0J2-06XW-00001
That one says the included power adapter is 2A, so check the WD support page linked above to see if it's enough for your drive. (and if you know how to solder, you can build your own very cheaply)
If that doesn't fix it, it may be that the drive isn't formatted properly. Reformat the drive to the whichever filesystem is needed by vita (probably Fat32).
and if that doesn't fix it either, then it may be a problem with the WD adapter hardware. In that case, you should open up the case, pull out the hard drive, and hook it up to a different adapter, like this:
https://www.newegg.com/p/36F-00MF-00019 (Note that this one is powered too, so you can throw away/return the powered USB hub linked above if you bought it.)