Could you tell us which is correct in the following sentence -
"have" or "has"?

Neither I nor my family have a financial interest or business relationship
which competes with or conflicts with the interest of the company.

It's "has".

There are two things to consider. The neither..nor construction governs a
singular verb if both elements are singular. In your example, one of the
elements is "family", which is a collective noun. This can be plural or
singular depending on the context. In your example, I judge your family to
be one entity; i.e. you are not considering all the individuals (e.g. (1) My
family is from London. (2) My family are living in different countries.)

Therefore, you have two singular elements in your neither...nor
construction, which should demands a verb conjugated singularly.