Yes this practice arises from the verse in Vishnu Purana which discusses the quality of food, once it is 'eaten'(?) by the Lord. The practice therefore has carried on since millennia.
anna-panadikam ca yat
nasti tad-bhaksane dvijah
yatha visnus tathaiva tat
bhaksane tad dvijatayah
tasman navartate punah
"O brahmanas, in the eating of food such as rice and water, etc.,
which has been tasted by the Lord of the universe [Vishnu obviously, since this is
Vishnu Purana] there is
no consideration of whether it is edible in inedible. Just as Lord
Vishnu is spiritual and is not subject to mundane transformation, so in
the same way His prasada is spiritual and eternal. Those
brahmanas, who consider that in the eating of Vishnu-prasada there is
some mundane transformation of its spiritual qualities and thus it can
become contaminated by contact with someone's mouth or hands, become
afflicted with leprosy, become bereft of their sons and wife, and go
to hell, never again to return from that place."
This is supported by Srimad Bhagavatam, which is said to be one the greatest book for the Vaishnavas, especially Gaudiya Vaishnavas who consider it even higher than Vishnu Purana, only second to Bhagavad Gita.
shadbhir masopavasais tu
yat phalam parikirtitam
bhunjatam tat kalau yuge
"In Kali-yuga they who eat Lord Vishnu's prasadam attain the same
result said to be gained by fasting for six months."