Are you the campground owner and/or do you have approval to do permanent electrical work?
Most campgrounds don't like people doing this sort of stuff because if the local inspectors see it and it was done without a permit, they can hassle the campground.
Now if you have approval to do permanent electrical work and the circuit isn't maxed out, just do a daisy chain off the first pedestal, not plugging in a 50 amp cord and running it thru a conduit.
One item, you didn't mention is what kind of distances are you talking about and how good is the existing power supply.
- If it's 20' away and there are no voltage issues, probably not going to be an issue.
- If it's 300' away and you've seen low voltage during peak periods, give a lot of thought to voltage drop. You may need to up-size the wiring or consider a voltage booster.