Canon suggests that goblins could certainly use wands, and probably other non-human creatures as well.
The Goblet of Fire quote you want is this one, which introduces the wand ban:
“Here, look.” Mr. Diggory held up a wand and showed it to Mr. Weasley. “Had it in her [Winky’s] hand. So that’s clause three of the Code of Wand Use broken, for a start. No non-human creature is permitted to carry or use a wand.”
This suggests that there are non-human creatures can use wands (although mastery is still up for debate). If they couldn’t use them, then the law would be redundant.
The wording of the law marks humans out as different from other creatures, and reads like a deliberate attempt to promote human supremacy. Various quotes from JKR suggest that you need a wand to do certain forms of advanced magic. Restricting access to wands would prevent them from using this magic.
In a 2001 interview, she suggests that wands are necessary for “really good spells”:
Do you need a wand to do magic?
You can do unfocused and uncontrolled magic without a wand (for instance when Harry blows up Aunt Marge) but to do really good spells, yes, you need a wand.
In a 2006 interview, she similarly says that wands are necessary to make potions:
There is a magical component to the potion, not just the ingredients. So, at some point you’re going to have to use a wand.
House-elves and goblins both have branches of advanced magic that don't require wands (apparition and metalwork, for example). However, neither show an aptitude in spells or potion-making; this does suggest that a wand ban was a deliberate attempt to prevent non-human creatures accessing certain branches of magic.
Case study: goblins
We have enough detail in the canon to study goblins in particular.
In Deathly Hallows, Griphook explicitly states that wizards are using a wand ban to restrict the magical abilities of goblins:
“The right to carry a wand,” said the goblin quietly, “has long been contested between wizards and goblins.”
“Well, goblins can do magic without wands,” said Ron.
“That is immaterial! Wizards refuse to share the secrets of wandlore with other magical beings, they deny us the possibility of extending our powers!”
[…]
“As the Dark Lord becomes ever more powerful, your race is set still more firmly above mine!”
This tension is hinted more subtley in Order of the Phoenix:
“It depends what they're [goblins are] offered,” said Lupin. “And I’m not talking about gold; if they’re offered freedoms we’ve been denying them for centuries they’re going to be tempted.”
Presumably these freedoms include the ability to carry a wand.
The “centuries” comment comes from the third WOMBAT (written by JKR), which states that the wand ban was passed in 1631:
The Ministry of Magic Decree of 1631, preventing all magical beings other than wizards carrying a wand.
There are various references to a goblin rebellion in 1612, based in Hogsmeade. There's no canon evidence that the wand ban was directly prompted by this rebellion, but it probably helped.
Finally, in the late 1990s, JKR wrote and released four copies of the Daily Prophet, called The Daily Prophet newsletters, to the UK fan club. I can't find full text copies online (see this discussion on HP Wikia), but the HPL Lexicon has a good summary, which includes this piece on goblins:
Goblin riots erupt in Chipping Clodbury
B.O.G. (Brotherhood of Goblins) supporters ran riot during a meeting with representatives from the Department for the Regulation and Control of Magical Creatures. The goblins used illegal wands for transfiguration and called for the release of activist Hodrod the Horny-Handed. Goblin-wizard relations are reported to be at an all time low.
Here we see the goblins actually using wands, and wizards getting angsty about it.
Given how keen goblins are to be allowed to use and study wands, it only seems logical that they would be able to use them. There’s nothing to indicate goblins are unusual in this respect, so we may assume that other non-human creatures can use wands as well.
