Is it permissible to dig a well on one's private property due to water scarcity and the negligence of municipalities, especially in Gaza where the well supplying the area is far away?

There is no objection to digging a well on your land for drinking and similar purposes, as the fundamental principle is the permissibility of an owner disposing of their property. This is unless it results in harm to a neighbor, or if the state restricts this permissible act to prevent a public harm, in which case the restriction must be observed.