No access quest.
With the new travel system, you should be able to ride the horses most of the way. Basically, in Lavastorm you start at the southern end of the zone and you have to get to the middle, but to do that you have to travel in a circle, going counter-clockwise. I haven't been to LS since the travel revamp, so I don't know if they'll drop you off at Sol's Eye or if you'll have to run some of the way. The worst part IMO is getting through the Temple of Sol Ro. I always forget the turns in there.