I think it is possible to apparate without a wand. May be wand makes it easer to apparate.
Ron was now trying to Disapparate without a wand.
“There’s no way out, Ron,” said Luna, watching his fruitless efforts. “The cellar is completely escape-proof. I tried, at first. Mr. Ollivander has been here for a long time, he’s tried everything.”
Harry Potter and the Deathly Hallows, Chapter 23. Malfoy Manor
If we go back to Harry Potter and the Half-Blood Prince, when the training for apparition test was going on. There is no mention of wand work.
He waved his wand. Old-fashioned wooden hoops instantly appeared on the floor in from of every student.
“The important things to remember when Apparating are the three D’s!” said Twycross. “Destination, Determination, Deliberation!
“Step one: fix your mind firmly upon the desired destination,” said Twycross. “In this case, the interior of your hoop. Kindly concentrate upon that destination now.”
“Step two,” said Twycross, “focus your determination to occupy the visualised space! Let your yearning to enter it flood from your mind to every particle of your body! “
“Step three,” called Twycross, “only when I give the command... turn on the spot, feeling your way into nothingness, moving with deliberation. On my command, now... one —”
Harry Potter and the Half-Blood Prince, Chapter 18, Birthday Surprises
Regarding Lily Potter's death, there might have been a charm (Anti-Disapparition Jinx) on the house like in Hogwarts as security feature. I say this because we don't see Voldemort apparating inside the house.
The gate creaked a little as he pushed it open, but James Potter did not hear. His white hand pulled out the wand beneath his cloak and pointed it at the door, which burst open.
He was over the threshold as James came sprinting into the hall. It was easy, too easy, he had not even picked up his wand
Harry Potter and the Deathly Hallows, Chapter 17. Bathilda’s Secret