Q:

Are hand motions required to use the Force? 

In almost every canon (that is, visual) source, Force users typically wave a hand to invoke the Force to move or manipulate objects, people and thoughts.  I get out-of-universe this is a visual cue that the Force is being used, but in-universe is it strictly necessary? 
(The only exception I can think of is when Luke is training on Dagobah and is balancing rocks while standing on one hand with Yoda on his foot.  He doesn't appear to be waving his hand to move the rocks.) 
Is this addressed anywhere in-universe, even in Legends? 

A:

No,
Hand motions are not required to use the force. 
The perfect example is in the Empire Strikes Back when Darth Vader faces off with Luke just before he cuts off Luke's Hand.  He's force throwing all sorts of stuff at Luke without even taking his hands of his lightsaber. 

There are many other examples of Darth Vader using the force without so much as making a move, like when he force choked Admiral Ozzel over communicae without so much as flinching.  All the while continuing his conversation with the newly promoted Admiral Piett.  
 
It appears that Jedi or Sith with advanced knowledge of the force can use it without any gestures of any kind.  

A:

No, they're not necessary.  But they act as a focusing aide and may be necessary for more difficult tasks.  Notice that no hand gestures were needed when Luke levitated C3PO:

But when he focuses deeply to attempt to lift his X-Wing, he extends his hand:

Similarly, as ShemSeger noted, Vader doesn't use hand gestures to throw (relatively) lightweight objects at Luke during their duel in the Empire Strikes Back.  However, when he attempts to match Obi-Wan's force-push in Revenge of the Sith, he makes this motion:

A:

During the Jedi Academy series of novels, Luke Skywalker talks to an independent tibanna gas miner on Bespin, named Streen.   They have a discussion about the Force and it's uses (Streen has been uncontrollably listening to people's thoughts/feelings, leading him to become a hermit).   Luke demonstrates his telekinesis by flicking his finger at a distant antenna and making it wave with a telekinetic strike. 
He encourages Streen to try as well, and tells him that the gesture isn't truly needed, though it can serve as a focus.   Streen, if I recall correctly, accomplishes the same task (with a weaker push) without the hand motion.   He claimed the hand motion had been distracting. 
It's no longer canon (thanks, Disney!)  but it was a good source for the answer to this question.