The correct word for filling out the application form And you can fill in a form because you're supplying missing information Fill out is generally used when you tell someone to enter all the fields on a form of more than one field Example: Please fill out this form Fill out means to complete by supplying requested information
Is it possible to use the verb fill with the word shortage? “The aid is intended to fill the food shortage in the area” But your example sentence is very strange, and not because of the choice of verb In “We can X each other’s shortages”, I can’t think of any verb that would make the sentence make sense without having to think up highly specific contexts