Both board and lodging have long histories of use independent of the other term, although in the modern day lodging is more commonly found on its own than board (in the relevant sense) is.
'Board' without 'lodging'
The use of board for "table set with food and drink" is rather old-fashioned, but examples of its use are not hard to find through Google Books searches. Here are three such instances that turned up when I searched for the phrase "sit at his board."
From Henry Summersett, Martyn of Fenrose; Or, The Wizard and the Sword: A Romance, volume 2 (1801):
He hailed the rich and the poor with the unanswered voice of friendship. He provided a feast, and opened his door, but no person stept over the threshold, or ventured to sit at, his board. I was provoked, and saw the infelicities attending a being who, actuated by pride or false sentiments, stood aloof instead of joining with his species.
From Anthony Trollope, The American Senator (1877):
And yet the hospitable hero who would fain treat his friends as he would be treated himself can hardly arrange his dinners according to the palates of his different guests; nor will he like, when strangers sit at his board, to put nothing better on his table than that cheaper wine with which needful economy induces him to solace himself when alone.
And from James McAllister & Grace Guerrant, Edward O. Guerrant: Apostle to the Southern Highlanders (1950) [combined snippets]:
He received me most graciously and welcomed me to his hospitable home near Wilmore, one of the most idyllic places I have ever seen. Here he showed himself the elegant gentleman he was, and to rest beneath his roof and to sit at his board has ever since been regarded as an outstanding privilege.
'Lodging' without 'board'
The term lodging occurs frequently in modern writing, unpaired with board. Here are three recent instances that a search for "offered" + "lodging" returned.
From Liz Carmack, Historic Hotels of Texas: A Traveler's Guide (2007):
Each of the featured hotels met three criteria: it is at least fifty years old; it originally opened as a hotel or, if not, became one and has operated as such for the majority of the time since the building was completed; and it offers lodging today.
From Touraj Daryaee, The Oxford Handbook of Iranian History (2012):
With travel relatively safe and unrestricted, wandering bands of qalandars (wandering, disreputable dervishes) became a familiar sight, while more traditionally minded Sufi khanqahs (religious hospices) offered lodging to more conventional travelers.
And from John Christopher, The Prince in Waiting (2015):
Tents were set up that night in the fields by Shidfield village. My father had been offered lodging but declined, on the grounds of not inconveniencing the villagers; but as he said, laughing in private, also because he had never yet encountered a village lodging that was not overrun with fleas, and polymuf giant fleas at that.
Alternatives to 'board and lodging'
One common way to refer to "board and lodging" (or "bed and board" or "room and board") is as "meals and a place to sleep." So in the situation described in the posted question, you might ask, "Will your aunt be providing meals and a place to sleep?" Of course, you might want to delve into the particulars—how many meals a day, whether the place to sleep is an actual bed in a private room, etc. But generally, I think, meals is a more apt synonym for board than food is.