What is confirmed: Mike Pence did have emotional moments in his public life,
including during his farewell remarks in January 2021 as he left office, when he held back tears while thanking supporters and colleagues.
That speech, however, was more of a political farewell than a personal revelation.
There are also other documented occasions where Pence reflected with emotion—such as in interviews and his memoir—particularly when discussing his relationship with Donald Trump after January 6 and the strain it caused.
The passage you shared seems to be more of a dramatic retelling or fictionalized account rather than a direct transcript from a real event.