Nope, they had no legal right.
A private person may arrest an offender if the offense is committed in his presence or within his immediate knowledge. If the offense is a felony and the offender is escaping or attempting to escape, a private person may arrest him upon reasonable and probable grounds of suspicion.
Making a citizens arrest is predicated on witnessing a
felony, or at least having immediate knowledge. As Arbery had not committed a felony, they had neither witnessed, nor had immediate knowledge, of him committing a felony.
This failure on their part to enforce a citizen's arrest is among the reasons I expect to see convictions in this case.