Are neurotransmitters part of the endocrine system?
Short answer
Synaptic signaling can be seen as a type of paracrine signaling, and is hence not an example of an endocrine system.  

Background
Khan Academy has a nice accessible overview on this confusing topic:

Paracrine signaling
Signalling between nearby cells that are near one another van occur through the release of chemical messengers that diffuse through the intercellular space. This relatively short-distance communication is called paracrine signaling. Notable paracrine signaling occurs during development, where growth-hormone gradients regulate the differentiation of the various tissues.  

Synaptic signaling
A unique type of paracrine signaling occurs between neuronal cells, namely synaptic signaling.

Autocrine signaling
Autocrine signaling refers to cell signaling to itself. Like paracrine signaling, autocrine signaling is important during development, but also in tumor metastasis.  

Endocrine signaling
Communication over longer distances often occurs through the circulatory system is called endocrine signaling. Often it occurs through the release of hormones into the bloodstream. Hormones are produced and released by specialized cells and target cells in distant parts of the body. Endocrine glands include the thyroid, hypothalamus, and the pituitary, among others. For example, the pituitary in the brain releases growth hormone that affects growth and other functions throughout the body.

Juxtracrine signaling
Cells can also interact via direct contact. For instance, gap junctions in animals and plasmodesmata in plants allow for a direct communication between cells by forming tiny channels, allowing small compounds and ions to cross the cell's membrane to a neighboring cell. It occurs for instance in the brain.