Yes you can test for hormone levels in a horse as in any mammal. But it doesn't necessarily tell you whether everything is working correctly as hormone levels change daily.
You can blood test to "prove" a rig - although in practice all that tells you is that the horse has more testosterone than it should; the most common cause of which is being a rig.
You blood test for ACTH levels to indicate Cushings / PPID and to monitor the effect of medication. ACTH is a hormone.
You can blood test for pregnancy - but only after the first 14 days. And for an equine ovarian tumour - although there is debate about how effective that actually is as some reports suggest over 50% of mares with palpable tumours also have normal hormone levels.