Harlequins number eight Alex Dombrandt is injured so Vunipola, who last played in the 2021 Six Nations, returns.
Scrum-half Care is named but England's most capped male player, Ben Youngs, does not travel with the squad.
The first Test is in Perth on 2 July, followed by matches in Brisbane on 9 July and Sydney on 16 July.
Care, 35, won the last of his 84 caps in 2018, but played for England in the 52-21 non-cap loss to 14-man Barbarians at Twickenham on Sunday.