Short answer
Neurotransmitter release is triggered by an influx of calcium ions from the extracellular space into the cytosol through voltage-gated calcium channels in the plasma membrane.
Long answer
Typical synaptic vesicle release is triggered by calcium influx through T-type calcium channels. These are voltage-sensitive channels in the plasma membrane; calcium flows through these channels from the extracellular space and is bound by synaptotagmin, a type/family of proteins associated with the vesicle fusion machinery (the SNAP/SNARE complex) to trigger (or, more precisely, to increase probability of) vesicle release.
Like just about every important class of proteins, these channels can be regulated by various mechanisms: gene expression, trafficking/colocalization, subunit composition, phosphorylation, etc. See the Iftinca & Zamponi review.
Calcium from internal stores, released by various mechanisms (calcium-induced, IP3-induced, can also be important to neuron plasticity and homeostasis but does not need to travel from the soma (ER extends throughout the cell). In some special cases, these internal calcium stores can be involved in release (see for example Babai et al 2010), but this example is from in photoreceptors in the retina which use a different release mechanism than typical neurons (they don't fire spikes, they have graded potentials and graded release probability, etc). Internal stores can also influence spontaneous release probability and may modulate release probability (see for example the review by Collin et al) but are not the primary "trigger" for release.
For more on the basic synaptic machinery, I'd recommend any edition (though of course recent is always best) of the textbook "Neuroscience" by Purves and coauthors.
Babai, N., Morgans, C. W., & Thoreson, W. B. (2010). Calcium-induced calcium release contributes to synaptic release from mouse rod photoreceptors. Neuroscience, 165(4), 1447-1456.
Collin, T., Marty, A., & Llano, I. (2005). Presynaptic calcium stores and synaptic transmission. Current opinion in neurobiology, 15(3), 275-281.
Iftinca, M. C., & Zamponi, G. W. (2009). Regulation of neuronal T-type calcium channels. Trends in pharmacological sciences, 30(1), 32-40.
Dale Purves, George J. Augustine, David Fitzpatrick, William C. Hall, Anthony-Samuel LaMantia, Richard D. Mooney, Michael L. Platt, Leonard E. White. (eds) (2018). Neuroscience. New York: Oxford University Press.