It seems to me that in the video there's a mix of two different stories: a famine during the expedition and the story of the 7 poor sahaba who wanted to support the expedition but had nothing but themselves to offer.
In sahih Muslim there's a long story which might be the one you are looking for:
During the time of Tabuk expedition, the (provisions) ran short and the men (of the army) suffered starvation; they said: Messenger of Allah, would you permit us to slay our camels? We would eat them and use their fat. The Messenger of Allah (ﷺ) said: Do as you please. He (the narrator) said: Then 'Umar came there and said: Messenger of Allah, if you do that (if you give your consent and the men begin to slay their camels), the riding animals would become short. But (I would suggest you to) summon them along with the provisions left with them Then invoke Allah's blessings on them (different items of the provisions) It is hoped Allah shall bless them. The Messenger of Allah replied in the affirmative. (the narrator) said: He called for a leather mat to be used as a table cloth and spread it out. Then he called people along with the remaining portions of their provisions. He (the narrator) said: Someone was coming with handful of mote, another was coming with a handful of dates, still another was coming with a portion of bread, till small quantities of these things were collected on the table cloth. He (the narrator said): Then the messenger of Allah invoked blessing (on them) and said: Fill your utensils with these provisions. He (the narrator) said: They filled their vessel to the brim with them, and no one amongst the army (which comprised of 30,000 persons) was left even with a single empty vessel. He (the narrator) aid: They ate to their fill, and there was still a surplus. Upon this the Messenger of Allah (ﷺ) remarked: I bear testimony that there is no god but Allah and I am the messenger of Allah. The man who meets his Lord without harboring any doubt about these two (truths) would never be kept away from Paradise. (Sahih Muslim and without explicitly quoting it was during the Tabuk expedition here)
One of the verses which is said to be quoting this expedition is (9:92):
Nor [is there blame] upon those who, when they came to you that you might give them mounts, you said, "I can find nothing for you to ride upon." They turned back while their eyes overflowed with tears out of grief that they could not find something to spend [for the cause of Allah ].
And the narrations which are related to this verse indeed include a story of a man who were poor and had nothing to give to help the Muslim expedition so he gave himself as sadaqah and was accepted their story is widely presented in tafsir books some of the narrations have been quoted by imam ibn Kathir with different narrations, at-Tabari (narration #17088 with his own narrator chain) and ibn abi Hatim (narrations #10200 and 10201 with his own narrator chains with different amounts of people), while al-Qurtobi presented in his tafsir (4th issue) the story as one out of many narrations which are declared as reasons for the revelation of this verse. Imam al-Haythami quoted a statement of 'Abdullah ibn Mughaffal saying he was among the people who were meant by the prophet's answer:
"I can find nothing for you to ride upon."
in his majma' az-Zawaaid (which was also quoted by al-Qurtobi) commenting that it was also by at-Tabarni via a good chain.