Vascular/phlebology/sclerotherapy: generally, how likely is it that an injection into leg spider veins accidently pierces a close artery? then what?

Very unlikely. : Very unlikely . Should that occur, probably the worst thing that would happen is a superficial ulceration in the skin, if you are asking about superficial spider veins.

Very Unlikely: Spider veins are so superficial and small that is is very unlikely that you would get anywhere near an artery doing sclerotherpy. This can be a worry when injecting deeper and larger veins & this is why ultrasound is utlized for deeper sclerotherapy. But spider veins are not deep & not large. If something like this happens, then a skin wound could form but have never seen it personally.

An actual artery: isn't likely to be injected. And, the solutions used to treat spider veins are weak and might not damage an artery. However, very small arteries - called "arterioles" - commonly communicate with spider veins. Arteriole pressure usually prevents the solution from entering, but if it does, a small wound ("cutaneous necrosis") may occur which heals very slowly. Good technique lowers this risk.