Continuing the story of Spider-Man without mentioning MCU history is not like ignoring it. New movies about Spider-Man will not refute anything that has ever happened. Viewers will still remember the MCU movies he has appeared in, and they will bring that knowledge to the theater. Just vaguely mention things, use character-based explanations to avoid issues related to continuity in the timeline, and don’t cause conflicts directly to the story of MCU.
Even without specific nouns, we will know who Peter, May and the rest of the characters are talking about. Prior to this, the series and the MCU were inseparable. The movie’s main character, Phil Coulson, appears alongside the MCU in Iron Man, chases the Skrulls in Captain Marvel, and is murdered by Loki in The Avengers. The MCU guest stars like Nick Fury also often “visit” this show. When Captain America: Winter Soldier revealed that HYDRA had merged with S.H.I.E.L.D, the show made a slight reboot and revealed the fact that the most significant character in the movie turned out to be a HYDRA agent. In season 2, S.H.I.E.L.D continues after events in Thor: The Dark World.