Does apple cider vinegar have apple cider in it?

Most likely in trace amounts, yes.

To make Apple cider vinegar, you start with Apple juice.

Yeast is added to the juice, which transforms the sugars in the juice to alcohol. During this time, care is taken to ensure that no oxygen is able to get to the "must".If it is allowed to completely ferment, nearly all (but unlikely completely all) off the sugarsa will be gone from the juice. At this point you have "Hard" cider (or just cider if you live outside the USA).

At this point, the airlocks are removed and a bacteria is introduced (or allowed to self-introduce) to the cider. This bacteria turns the alcohol into acedic acid- which is the stuff that gives vinegar its strong taste and smell.

It is unlikely that there would be a complete transformation of all the alcohol, so yes there is most likely alcohol in the vinegar.

However, it is in amounts so small as to be nearly impossible to detect.