In the Belgariad (and other works by Eddings within the same universe), is it always necessary to use spoken words to cast magic through the Will and the Word? Or is it possible to utilize it purely by will alone, and perhaps simply thinking the word?
If we stuck purely to the Belgariad, then not necessarily, as we see times where Belgarath and Polgara do things using only their Will. Moments where a Word was used were often used as dramatic effect (such as Belgarath asking a tree to bloom and ordering the knight to tend to it). Often times it is used when they were in distress, or when being dramatic. Take every example where they change into an animal, they never say "Wolf" or "Change". Even the battle of Chutchik and Belgarath, there are no words, just raw energy and will surging.
Outside the Belgariad, no it's definitely not necessary. During the battle of Vo Mimbre, you see the other sorcerers constantly doing their thing without as much as a peep coming out of them. Another example of no words being used would be the time where Belgarath first meets Beldin. The dwarf translocates several tons of stone to the base of Belgarath's tower without saying a word.
However, your question title and question body ask about two separate things, which I've noticed two years later.
For Sorcery, the word isn't as important, and can be done silently.
For Magic, the opposite is true. Magic in this world being the ability to summon and control demons. This is implemented by the Morindim and the Karands. The words are incantations that are needed to control them and force them into what you imagine them to look like. If the magician botches this up, the demon is freed and bad things happen.
Yes, a word is necessary to complete any sorcerous effect cast by a user of the Will and the Word. In the Belgariad series, practitioners of sorcery (called The Will and the Word) needed both to gather their will and then speak whatever word they used to trigger the spell effect. The word was less important than the will, however.
From the first book of the Belgariad, page 163, in a conversation between Belgarath the Sorcerer (Mr. Wolf) and Garion:
"The Will and the Word," Wolf said, his long cloak whipping about him in the stiff breeze. "It isn't difficult."
"I don't understand," Garion said.
"You simply will something to happen," the old man said, "and then speak the word. If your will's strong enough, it happens."
"That's all there is to it?" Garion asked, a little disappointed.
"That's all," Wolf said.
"Is the word a magic word?"
Wolf laughed, looking out at the sun glittering sharply on the winter sea. "No," he said. "There aren't any magic words. Some people think so, but they're wrong. Grolims use strange words, but that's not really necessary. Any word will do the job. It's the Will that's important, not the Word. The Word's just a channel for the Will."
"I always thought that sorcery had to be done with long spells and strange signs and things like that," Garion said.
"Those are just the devices of tricksters and charlatans," Wolf said. "They make a fine show and impress and frighten simple people, but spells and incantations have nothing to do with the real thing, It's all in the Will. Focus the Will and speak the Word, and it happens. Sometimes a gesture of sorts helps, but it isn't really necessary. Your Aunt has always seemed to want to gesture when she makes something happen. I've been trying to break her of that habit for hundreds of years now."