Walter Mendoza, 30, was exasperated with his mother, Ana, as they walked in to Supremo Foods in Allentown, Pa., last week.

Their shopping list had to be kept to a minimum, said Mr. Mendoza, angrily pointing the blame at President Biden: “Because your president got these prices this high!” he told his mother.

“Because your president messed up big time,” Ms. Mendoza, a 52-year-old warehouse manager, retorted to her Trump-supporting son. “Inflation came after the pandemic that he screwed up, bro.”

Arguments about prices and presidents are taking place across America, but the Mendozas’ dispute comes with extraordinary stakes.

The 2024 campaign has marked an arrival of sorts for the nation’s roughly 36 million eligible Latino voters — a group now so large, geographically dispersed and politically divided that it will be crucial in deciding who wins the White House.