After reaching friendship level 7 with Woody, talk to him to start the Wanted: Aliens quest. You also need to have progressed your friendship with Scar in order to be able to enter his Cave which will be needed in the quest. Follow these steps to complete the quest:
Gather the materials for the new Claw Machine
Craft the new Claw Machine
Meet with Woody inside the Carousel and bring the Claw Machine to him
Talk to Buzz
Find the Alien Toys and give them to Woody
Step 1: Gather the materials for the new Claw Machine
Woody will give you some of the old Claw Machine’s parts. You’ll then need to gather the following materials needed to fix the Claw Machine:
x2 Iron Ingot
Step 2: Craft the new Claw Machine
Once you have all the materials, head over to a crafting station to craft the new Claw Machine.
Step 3: Meet with Woody inside the Carousel and bring the Claw Machine to him
Head inside the Carousel and then give the Claw Machine back to Woody. He will then ask for your help in locating the Aliens.
Step 4: Talk to Buzz
Go to Buzz and he will tell you the locations of where the Aliens have landed.
Step 5: Find the Alien Toys and give them to Woody
You then have to go to the locations where Buzz have seen the Aliens:
Under Dazzle Beach’s sand
Frozen in ice near a waterfall
In the ground near the Giant Willow
In Scar’s Cave
Once you’ve gathered all of them, go back to Woody and give him his Alien friends back to complete the quest.
After completing the quest, you can then continue working on leveling up your friendship level with Woody.