Certainly most decent hotels could either help you organise one, or let you know where to.
The distance is around 200km, and a shared taxi would be a reasonably economical way of doing it.
Prices vary, but a quick google search for 'taxi manali ladakh' is very easy to find a bunch of sites that let you organise them, for example - Discoverledahk.in - they have a phone and email contact details. I suspect, however, that it'd be cheaper to organise once you get there, based on experience travelling in the area and other countries that have shared taxis.