There are several reports in literature of software based vessel morphometry determination.
There are also skeletonization and other similar algorithms written for the freely available NIH - ImageJ software that will help. These mostly evaluate the entire vascular tree, but you could write specific code in softwares like Matlab or adapt published models from ImageJ to do the start and end point based thickness detection to suit your needs.
In order to measure thickness of vessels, you need to know: what are vessels? So, first the segmentation must be done. The image is not illuminated homogeneously; I would start with correcting for that...
Finding a proper threshold is difficult, especially because the vessels seem only 1 to 3 pixels thick: the error introduced by choosing a threshold that nicely separates vessels from background, but on the other hand 'skins' the vessels, is huge!
Segmentation should be done first followed by calculating the thickness of main vessels automatically. If you are interested in specific part of vessels, you may use labelling followed by measuring the thickness.
I would recommend these two articles for better segmentation of retina images: