"Relationship to" vs. "Relationship with"

I think that the answer would depend on whether you are referring to how people/things are connected, in which case "relationship to" would be better, or how one would qualify or categorize the relationship, in which case "relationship with" would be better.

Ideally, the choice would not be between those two terms, but rather between "relation to" and "relationship with". See "Relation" vs. "Relationship".

A relationship between two people A and B is the set of connections -- familial, social, commercial, etc. -- between them. The relationship of person A to person B means those connections from the one-way point of view of person A. In filling out a form, this is almost always a question about familial connection:

Q: What is (person B's) relationship to the applicant (person A)?
A: Person B is person A's father.

In other words person B's relationship to person A is that of male parent. Usually, this may be turned around from the point of view of person A:

Person A is person B's son.

That is person A's relationship to person B is that of male offspring.

Relationship with usually asks for the broader mutual connections between the two parties:

Q: What is (person B's) relationship with the applicant (person A)?
A: Person B and person A have been business partners for three years. They jointly own three commercial buildings, and Person A has sponsored person B for club membership.

Whether this distinction obtains for this particular application cannot be discerned without having the application posted.