J.K. Rowling has stated on her website that the Summoning Charm only works on inanimate objects, explaining that creatures and humans can only be summoned by summoning objects they are holding or wearing.
‘Accio’ only works on inanimate objects. While people or creatures may be indirectly moved by ‘Accio-ing’ objects that they are wearing or holding, this carries all kinds of risks because of the likelihood of injury to the person or beast attached to an object travelling at close to the speed of light.
Welcome to my new website! - J.K. Rowling
Wonderbook: Book of Spells, a PS3 game generally considered to be canon, contains this quote supporting Rowling's statement.
Once you have mastered this charm, you have the ability to Summon any object of which you have need (Providing, of course, that you respect the International Statute of Wizarding Secrecy since no object should be Summoned if it is likely to whizz under the nose of a Muggle). The Summoning Charm is ineffective on most living creatures, and those that can be Summoned are rarely worth Summoning (such as Flobberworms).
Wonderbook: Book of Spells (ellipses replaced with spaces), see source video
However, a story told earlier in the game seemingly contradicts this, stating that Gideon Flatworthy, a man who was apparently rather skilled in the Summoning Charm, managed to summon the cows that crushed him to death.
Experts still debate what precisely led to his final, foolish act. Most agree that he had probably become delirious through lack of food although some think that he was determined to die as he had lived. All we know is that, on the eighteenth of September, 1743 Flatworthy attempted to Summon himself an entire farm complete with livestock, cosy cottage and well-stocked larder. Naturally, the buildings would not shift, but the furious farmer followed his flying cows to the cave on the hill and discovered Flatworthy, still lying on his cushions, but crushed to death beneath a pile of hay bales and cattle.
Wonderbook: Book of Spells (ellipses replaced with spaces), see source video
While it's possible that the story was exaggerated or fictional, it is billed by fact, and was apparently the subject of dissenting opinions among experts. It's also possible that cows are one of the few animals that can be summoned (though this seems unlikely as they are non-magical and not particularly uncommon) or the cows in question were wearing clothes at the time. However, this is not the only example of the Summoning Charm being used on animate objects, both in likely-fictional stories and in reality. Examples of animals being directly summoned magically, in chronological order:
Inside one house the little girl's warts vanished as she slept; the lost donkey was Summoned from a distant briar patch and set down softly in its stable; the sick baby was doused in dittany and woke, well and rosy.
The Tales of Beedle The Bard - page 10 - "The Wizard and the Hopping Pot"
When the poor little Snidget flew my way I did a Summoning Charm. You know how good my Summoning Charms are, Pru – of course it was easier for me to aim properly, not being mounted on a broomstick at the time. The little bird came zooming into my hand. I stuffed it down the front of my robes and ran like fury. Well, they caught me, but not before I’d got out of the crowds and released the Snidget.
Quiddich Through The Ages, pages 12 & 13 (in the writing of Madam Modesty Rabnott)
NEWT: Accio Niffler!
The case bursts open and a Niffler jumps out.
Fantastic Beasts: The Crimes of Grindelwald - The Original Screenplay - scene 48 (reformatted slightly)
NEWT (of the Niffler): Where is he? Ah, Accio Niffler.
The Niffler is carried by the spell back into the case. NEWT takes the case and dashes off.
Fantastic Beasts: The Crimes of Grindelwald - The Original Screenplay - scene 48 (reformatted slightly)
Harry pointed his wand at the bullfrog that had been hopping hopefully towards the other side of the table – 'Accio!' – and it zoomed gloomily back into his hand.
Harry Potter and the Order of the Phoenix - chapter 18
"Reckon there's salmon, or is it too early in the season? Accio Salmon!"
There were several distinct splashes and then the slapping sounds of fish against flesh. Somebody grunted appreciatively.
Harry Potter and the Deathly Hallows - chapter 15
There are a few other instances of animals being summoned indirectly or unsuccessfully, such as Newt summoning the Niffler by summoning the jewelry in its pouch, or Harry failing to summon Hagrid and Hedwig during the Battle of the Seven Potters (which I think we can put down to Harry's inexperience with the charm; even when he summoned his Firebolt, he had been practicing just before he managed it, and the charm gets harder with range). I'd also give an honorary mention to Ron for "Accio Brain!".
But, what gives? Is there any possible explanation for the inconsistency in the ability of this charm to summon animals?
