2014 rulebook page 18:

When a power or ability indicates that the character “may” do something, that indicates an option that the player of that character can make when the situation presents itself. Other than that, powers and abilities are not optional. A special power that says “Character can use Blades/Claws/Fangs and Stealth” does not allow the character to choose whether or not it is using Stealth (since that standard power’s description does not use the word “may”) but does allow the character to choose when it will roll a d6 for its close combat attack (since the description of that standard power does use the word “may”).

Based on this, the fact that Splitlip's power doesn't say that a character "may" make a relic roll as a free action, all relic rolls made when Splitlip is on the team should be free actions. There should be no way to give the character a power action to make a relic roll.

I can't argue with that, but perhaps Chris has already asked and the intent was for Splitlip to be optional. The inclusion of the word CAN certainly makes it seem that way, otherwise why put it there?

Would've worked just as well without the word "can" and avoided any confusion.

"Friendly characters use free actions instead of power actions to make relic rolls and modify their rolls by +1 if not already modified by this effect."

This is not true. The removal from the Book is not the relic roll. The removal of a hammer from the Book is either a free action, at the beginning of your turn, or a power action, any other time during your turn.

After the hammer has been removed from the Book and placed in the square, and after choosing to use the free Quake, you then, without Splitlip, give the figure in the square a power action to pick it up as an automatic success. Splitlip turns the power action of the relic roll into a free action.