A neutral object has no net charge since its positive and negative charges cancel each other. However, I’m wondering how, in terms of fields, they do cancel each other. Since fields are vectors, won’t the spatial arrangements of the charges affect how the fields cancel? How exactly do the fields created by electrons or protons cancel in a neutral object?
Each individual atom is presumably neutral as well. So, we can simplify the model of a neutral object to a neutral atom and then go from there. Each atom has a positive core and a negative cloud of electrons around it as I'm sure you are aware. The electrons on average have a uniform field emitted in all directions and so does the nucleus. Because the electrons spend time one all sides of the nucleus it can be modeled as if they were acting from the nucleus. This is similar to how we model the Earth as a point particle when calculating its orbit around the Sun. If you consider the electrons and protons to essentially be in the same location with identical charge then it's as if neither is there at all (electrically). Keep in mind this is an approximation to reality and a classical interpretation of electromagnetism.