These are very similar, and they all can mean the same thing.
Offer and volunteer are closest, but volunteer is usually more formal than offer. There is usually a selection process for choosing a volunteer to fill a specific role, or at least a well-defined need. Both of these come initially from an external source, and must be approved by someone else. You may volunteer to help, but someone in charge could refuse, perhaps selecting someone else to help. An offer may also be refused.
"Of your/my/her own accord" comes from within you. You are the one who decides if you will do the task.