By my estimation, a 5-character party is needed:
High Elf / Druid / Sage: Common, Elvish, Druidic, +3 of player's choice
Vedalken / Knowledge Cleric / Acolyte: Common, Vedalken, +5 of player's choice(!)
Goblin / Rogue / Orzhov Representative: Common, Goblin, Thieves' Cant, +2 of player's choice
Tiefling / Storm Sorcerer / Simic Scientist: Common, Infernal, Primordial, +2 of player's choice
Deep Gnome / Ranger / Azorius Functionary: Common, Gnomish, Undercommon, +3 of player's choice
This uses races, classes and backgrounds that I collated from the Player's Handbook, Guildmaster's Guide to Ravnica, Volo's Guide to Monsters, Xanathar's Guide to Everything, and the Elemental Evil Player's Companion.
This gives the party a total of 25 languages understood fully (reading, writing and speaking), including Thieves' Cant and Druidic, which covers (assuming no players take duplicate chosen languages) all 25 languages found in those books.
*EEPC adds the Aarakocra language and the Deep Gnome subrace.