The phrase appears in the VeggieTales episode, titled Madame Blueberry, which aired on July 21st, 1998.
The episode presents the three sales scallions approaching Madame Blueberry in her house and attempting to lure her into buying their products by singing a song for her, with the initial line “Allow us to introduce ourselves”.
On January 10th, 2017 the song was uploaded to the YouTube channel Classic VeggieTales Fan, fitted with lyrics and amassing great popularity shortly after its upload.
The scene was turned into a meme about middle aged Indian men approaching teenage Instagram models, fitted with the macro “Allow us to introduce ourselves”.
The meme was initially uploaded to Memedroid and the next day it was already present on Reddit.
Following this, the format was already established and memers across all platforms started producing and uploading new jokes, using the macro from VeggieTales.