That's a difficult one. I do not think that the cells release different factors (or it is not known) dependent on vascular sprouting or new vessel formation. On endothelial cells you can label for integrin alpha v-beta 3, an integrin upregulated in endothelial cells undergoing neoangiogenesis. This one is not secreted. Good Luck to you!
An alternative is to use a specific marker for vessels (CD31, D240-Lymphatic) and analyze the vascular density (count) comparing to see if there is any increase.
I agree with Andrew, if you want to assess neoangiogenesis in blood samples, the strategy should be to go for soluble growth factors. VEGF is the best known GF for angiogenesis and there are plenty of ELISAs around. Another indirect option may be to analyze endothelial microparticles in FACS. Good luck!
If you are doing an animal experiment and are going to be imaging vessels, you can treat the animal with either BrdU or EdU (invitrogen) for a couple of hours before sacrifice to label cells that undergo S-phase during that time. Then label your tissue with Pecam-1 or your favorite endothelial cell marker and anti-BrdU or use the EdU detection kit.
Circulating VEGF definitely DOES NOT correlate with angiogenesis as demonstrated in both animal and human studies. It has too short a half-life (being largely taken up by platelets) and is produced for functions other than angiogensis (e.g. in lung and kidney) in sufficient quantity that signal/noise would likely be a problem in any event. There have been reports of a correlation between increased VEGF levels and efficacy of anti-VEGF therapy but the correlation isn't strong enough to serve as a pharmacodynamic marker. Right now identification of a good biomarker for angiogenic activity is an area of intense ongoing investigation. In biopsies, the suggestions given above (endoglin, alpha-v beta-3, pericyte coverage, etc.) are good ones. However, given that you're asking about blood markers, unfortunately there aren't any currently known.
Expression of AlphavBeta3 Integrin seems to be fairly specific for neoangiogenesis. For human samples you could use LM609 or a similar antibody to localize new capillary blood vessels
VEGF release is not specific for neoangiogenesis but also for vasculogenesis (expanding already exisiting collateral vessels and sprouting of existing vessels in contrast to new vessel formation)
Circulating Endothelial Cells (CEC) in blood sample has been used as a marker of Tumor Neo-angiogenesis and Tumor Growth in clinical studies. It has been shown the correlation between these CEC and growth factors which are well known to increase in peripheral blood of patient with cancer. You can perfectly find in the literature the methods for enumerating these CEC in blood samples and use this result as an expression of new blood vessel formation.