Fennekin and Squirtle
They make a good combo for the big battles.
Unfortunately, neither Delphox or Blastoise will be too much help with them, besides Blastoise having Water STAB against Flareon. Delphox's STAB Psychic moves will be resisted by Meowstic, and Absol/Greninja are immune.
Delphox's STAB will always be good against her Chesnaught. Otherwise, her Delcatty will die to anything, and her Goodra doesn't resist Psychic.
In the first battle he uses a Corphish which neither Delphox or Blastoise can do much to. Later on in the third battle though, Blastoise can defeat his Talonflame, and Delphox his Roserade.
In your first battle with him (which is a multi-battle) Blastoise and Delpox won't perform outstandingly. However in the last battle, Blastoise can destroy his Aerodactyl, and Delphox resists Florges' STAB attacks.
Bug type gym leader Viola. Fennekin's STAB fire moves will be supereffective.
A Rock type gym, lead by Grant. Wartortle's Water moves will be very helpful here.
A Fighting gym. Braixen learns Psybeam at level 17. It's not quite STAB yet, but will take care of Korrina's team with ease.
Grass type gym lead by Ramos. Braixen learns Fire Spin at level 22, and alternatively Delphox learns Flamethrower at level 41.
Delphox isn't weak to Electric, and Flamethrower can take care of Magneton. Keep Blastoise out of this one though.
Here come the Fairies. Fire resists Fairy and Fairy is neutral to Psychic. You can get some powerful neutral hits here from both Delphox and Blastoise, and Blastoise can learn Flash Cannon through TM.
More powerful neutral hits from Blastoise. Delphox can learn Shadow Ball through TM, but watch out for Slowking especially.
Delphox's STAB Fire moves will sweep Wulfric's team. And if Delphox does fall, Blastoise resists Ice.
Steel type Elite Four. Delphox can do some damage, but be wary of Aegislash and Probopass.
Fire type Elite Four. Blastoise has supereffective STAB.
Dragon type Elite Four. Delphox can take care of Dragalge with Psychic, and Blatoise can learn Ice Beam through TM.
Water type Elite Four. Keep Delphox away from this battle, and Blastoise can serve as a wall if need be.
Delphox can take care of Hawlucha, and Blastoise can defeat Aurorus and possibly Tyrantrum. Gourgeist can be beaten by Delphox; but watch out for it's Ghost moves.
Delphox has supereffective STAB against his Mienfoo/Mienshao. Blastoise can beat his Pyroar.
Delphox's and Blastoise's types compliment each other. Delphox resists Grass, and Blastoise resists Water. Delphox is a speedy special attacker and Blastoise is a great wall. Together they make a fantastic pair that can help your team through the whole game. I do recommend having a Fairy type to fill in the gaps, if you can get one. Charizard and Venusaur are weak to many more Elite Four members/Gym Leaders, so Blastoise is the best option. (Thanks to Jason the Sly Nicvee for move suggestions!)