In general, no
Most campsites are owned by someone, whether it's a private owner or some organisation. You can't just randomly dig up someone else's property without explicit permission from them.
Many campsites also are on grassed fields, especially in Europe. Covering grass with soil basically destroys the grass, leaving an area of bare soil which will be taken over by weeds. Future campers certainly will not thank you for turning a camping field into a thistle patch.
Yes I know top-dressing is a thing for lawns, but levelling a site is well past this. In any case, top-dressing normally uses a mix of sand and soil to improve drainage; you can only use plain soil if the soil is sandy or very light loam. Were you thinking of testing the soil structure before doing this levelling? I would seriously doubt it.
Competent gardeners who need to carry out levelling of a grassed site will cut away the turf, scrape off the topsoil, level the subsoil, then replace the topsoil (in an even layer), and finally replace the turf on top. This certainly is not a quick process. It also requires you to stay off the turf for some time after levelling, so that the grass can recover. It should be clear that this is not something you can do during a camping trip.
Maybe if you can guarantee no ill effects
If the ground is entirely bare (maybe in woods or in desert conditions), and you are camping informally on somewhere in the wilderness which does not have obvious owners/users, and you know that no-one else is likely to be camping there after you, then maybe you can get away with it. In that case, whatever you do is not going to adversely affect the area or other people.
But this is pretty unlikely. And even on bare ground (as @Yakk says in comments) you may find that you're hitting roots when you dig out soil for levelling.