For pirates in the Golden Age of Piracy, specifically the 1710s and 1720s, practically all ships and missions were "self financed" to use the terminology in the question. Most ships were stolen, a few were owned by people involved (that mostly applies to crews beginning from scratch and not crews formed by splitting existing one). Supplies were a combination of stolen and bought via stolen goods/money in shady ports/black market operations. There were no banks or money lenders for pirates.

If you want to see serious financing of commerce raiding operations, you'll want to look into privateering operations of the era. Although, sometimes operations that officially started as privateering operations (or even merchant operations) ended up becoming pirates later on (see the case of Captain Kidd).