Is it possible to remove a coaster brake from a preschooler's bike? My boss bought her 4 and 3 year old a bicycle for Christmas, but she didn't notice that one had double hand brakes and the other has only a front hand brake and a coaster brake in the back.

The 4 year old is the one who has the bike with coaster brake and he can't get the hang of riding the bike because every time he goes to push he just engages the coaster brake instead.

We've been looking for bikes that have no coaster brakes but all of the ones his size have them or they very expensive. We found some with double hand brakes but they still have the coaster brake.

Would it be possible to get one that has double hand brake and coaster and remove the coaster brake completely from it?
Short answer - no, not easily.

Kids don't have the hand strength to use handbrakes properly, so a coaster brake is going to function better, once he gets the hang of it.  

He's activating the brake by pedalling backwards.  So don't do that.  Instead encourage him to push off with pedals level instead of optimal angle.  That will discourage the rear brake from kicking in.

If you do want to remove it, you're better off finding another rear wheel that fits and lacks the coaster hub.  The local recycling shop here has plenty of bikes for $10, great parts donors.

Can Mister Four ride a 2 wheeler or is he on training wheels?  Has he had time on a balance bike?  He might benefit from "scootering about" on a two wheeler with no pedals first.