Second row Will Rowlands, who is set to arrive at Rodney Parade this summer, and former Dragons Cory Hill, Hallam Amos and Nick Tompkins have also made the squad.
Carter and Basham are two of five uncapped players in Pivac’s party along with forward Gareth Thomas and backs Ben Thomas and Tom Rogers.
Scarlets centre Jonathan Davies is Pivac’s captain, while regional team-mate Leigh Halfpenny needs one more appearance to reach a century of Test caps.
The reigning Guinness Six Nations champions host Kingsley Jones’ Canada at Principality Stadium on July 3, before back-to-back home matches against the Pumas.
Former Wales U18 and U19 captain Carter is included in the party at the end of his first season playing under Dean Ryan.
The 20-year-old second row, from Caldicot, made his Dragons debut in the Guinness PRO14 clash with Munster last November and earned a first start at Glasgow the following month.
While uncapped, Talywain’s Basham, 21, who returned to action in April after fracturing his forearm on New Year’s Day, has featured in Wales squads before.
”We are really looking forward to this summer, getting to work with this squad and the three home Tests that we will face,” said Wales head coach Pivac.
”It is great that we are able to select five uncapped players in the squad.
“It is an opportunity, much like the autumn campaign last year, to get these players into the environment and to expose them to Test match preparation and in some cases Test match rugby.
”This summer isn’t just about the uncapped players, it is also about internationals gaining more experience.
“It is a chance for them to get more starts and appearances under their belt and to step into leadership roles, so it is going to be an important period for us.
”We will have a training camp in North Wales before returning to Cardiff for our three Tests, so it is a great block of time we have together.
”Whilst the Six Nations success is still fresh in many people’s memories, it is important we continue to build both depth and our game as we continue to the World Cup.”
Rhodri Jones (Ospreys) (20 Caps), Nicky Smith (Ospreys) (39 Caps), Gareth Thomas (Ospreys) (*Uncapped), Elliot Dee (Dragons) (37 Caps), Ryan Elias (Scarlets) (17 Caps), Sam Parry (Ospreys) (4 Caps), Leon Brown (Dragons) (17 Caps), Tomas Francis (Exeter Chiefs) (57 Caps), Dillon Lewis (Cardiff Rugby) (28 Caps), Adam Beard (Ospreys) (25 Caps), Ben Carter (Dragons) (*Uncapped), Cory Hill (Cardiff Rugby) (32 Caps), Will Rowlands (Wasps) (7 Caps), Taine Basham (Dragons) (*Uncapped), James Botham (Cardiff Rugby) (6 Caps), Ross Moriarty (Dragons) (45 Caps), Josh Navidi (Cardiff Rugby) (28 Caps), Josh Turnbull (Cardiff Rugby) (10 Caps), Aaron Wainwright (Dragons) (29 Caps).
Kieran Hardy (Scarlets) (4 Caps), Tomos Williams (Cardiff Rugby) (22 Caps), Rhodri Williams (Dragons) (3 Caps), Callum Sheedy (Bristol Bears) (9 Caps), Jarrod Evans (Cardiff Rugby) (6 Caps), Jonathan Davies (Scarlets) (88 Caps) (CAPT), Willis Halaholo (Cardiff Rugby) (4 Caps), Nick Tompkins (Saracens) (10 Caps), Ben Thomas (Cardiff Rugby) (*Uncapped), Hallam Amos (Cardiff Rugby) (23 Caps), Leigh Halfpenny (Scarlets) (95 Caps), Jonah Holmes (Dragons) (5 Caps), Owen Lane (Cardiff Rugby) (2 Caps), Ioan Lloyd (Bristol Bears) (2 Caps), Tom Rogers (Scarlets) (*Uncapped).