Is it possible to play Child of Light without a mouse?

This would depend on your mouse pad skills. Personally I can play it with ease, please keep in mind that this is also more of a keyboard game then "Point and Click". The game also has partial controller support allowing you if you have a controller use that instead.

Edit: The game is more than 95% Keyboard based.

Though it is possible to play Child of Light with just keyboard controls, it is much better to play with both keyboard and some mouse/touchpad control. Most of the game doesn't require much mouse dexterity, so using a touchpad instead of a mouse would not be much of a disadvantage.

In the overworld, Aurora's movement and interactions are controlled through keyboard alone. It is possible to complete all actions without the mouse. However, with the mouse, you can open some chests remotely and reveal certain areas without Aurora physically entering them (thus reducing the risk of running into hazards such as spikes), heal Aurora slowly and stun enemies. Most of these can be accomplished with Aurora stationary, so you can even play with only one hand alternating between tocuhpad/mouse and keyboard.

In battle, all of the characters actions are chosen by keyboard. With a mouse, you can heal friendly characters and slow down enemy creature's progress on the timeline, and thus making combat much more strategic. Since slowing enemies only takes place between turns, and healing can be done when the game is paused while you make decisions, it is again possible to play with only one hand.

In short, though the game is mainly played with keyboard, playing without a mouse at all would make the game much more boring. The game doesn't require much precision or speed in mouse movements, so you won't lose much by playing with a touchpad (though there are ergonomics concerns).