As others have said you  would need to check with your site operator about the use of an external  gas cylinder, and storage of same when your touring. Not  all sites will have the same arrangements as they are  independent  companies.
Regards to using  an external  cylinder with a touring  caravan,  you  may  have problems with finding  a high pressure pigtail ( the connection pipe with the correct  couplings from the cylinder to the gas pressure regulator built  into the caravan) that  is long  enough to  reach  and external cylinder.
But  apart  from that  the gas should work correctly.  
Others have alluded to the dangers of transporting gas cylinders effectively  loose inside a car .  That  is definitely  a NO!   Just consider some facts. LPG cylinders for caravans are "Vapour Take Off cylinders" This means the gas valve on the cylinder must  ALWAYS  be at the top of the cylinder, to  ensure it  only allows gas vapour to be released. If a cylinder falls onto its side that  would mean the valve would no only  be vertically  half way  up the cylinder,  and it  could be immersed in the Liquefied gas content rather than the vapour.
In the event the valve is  not  fully  closed or it  has been damaged, instead of just leaking  vapour, which is serious enough on its own, the prone cylinder would be leaking liquefied gas.  In the process of filling an cylinder of LPG the pressure is sufficient  to  compress the gas to the point  it  condenses into its liquefied phase. That is roughly a compression  ratio of 260 times. If you  leak  liquefied gas that  liquid will instantly  try to  boil and convert is self into  its vapour phase which  means it  expands by about  260 times.  so  a small liquid spill becomes a much  bigger  gaseous cloud.  And in the process of expanding it draws the energy it  needs to  vapourise from the atmosphere around it. Is temperature will drop enough that  if you  get  liquefied Propane on your skin it  will  cause a serious frost  bight.
Then there is the risk of having a highly  combustible fuel lose inside an enclosed space such as a car.  Pure gas will not  burn becasue it needs to  mix with  air to  gain  access to oxygen. Broadly  speaking  you  need a ratio  of gas to air in the range of 4 to  15% to  support  combustion.  it therefore does not much of a spillage to  produce a combustible mixture inside a car.  Also  bear in mind there are  plenty  of electrical sources of ignition within a car. It the mixture ratio optimum not  only  will the  mixture burn it  might  have the flame speed to  be considered an explosive with the associated force to  physically  blow out  windows,  not  to  mention the concussion injuries such an event  might  impact  on  anyone in the car.
There is a legal  requirement for vapour take of cylinders to  kept upright,  and to  be secured to prevent them falling or otherwise moved,  especially when being transported.
When towing your caravan the cylinder(s) must  be secured in the gas cylinder locker, and turned off at the gas bottle or disconnected.