Touring sites next to train stations

Will a heritage railway do? Bishops Lydeard Camping and Caravanning is just five minutes walk from station on the West Somerset railway with steam trains to the coast at Minehead.

https://www.campingandcaravanningclub.co.uk/campsites/public-transport/

https://www.ukcampsite.co.uk/articles/view.asp?id=580

You'll have to do a bit of work yourself. From memory there were quite a few sites near Dawlish that would suit.

Whilst there are sites within walking distance of Stations there are far more if you are prepared to use a short bus ride to get to the station. I think also you need to say how far you are prepared to walk which might depend on how your mobility is.

The C&CC site at Moreton in Dorset the station is just across the road, there is also a CMC just a little further along the road. Lady's Mile at Dawlish Warren is walkable to the station although its a steep walk back, but you can get the bus. 

David

Minnows Touring Park (formerly an AS) near Tiverton is not far from Tiverton Parkway Station.

Moreton in the Marsh Club Site is around 15-20 mins walk from site gates to station at the North end of the town. Level walking, on good paths. No idea where Cotswold Line runs to and from, but think it’s main line.

Of course there are city/town sites that would satisfy your request, such as Bristol or Brighton. 

For town sites there is also Salisbury C&CC site about 40 minutes walk to the station. It really does depend on how far the OP wants to walk. Brighton, mentioned above, is about an hour from the CAMC site, Bristol Temple Meads 35 minutes from Baltic Wharf. However, buses that reduce the walked distance significantly are available at all of them.

Club site at Crossways is adjacent to the railway line. C&CC have a site the other side of it.

The station is inbetween the 2. Trains go to Weymouth, not sure of other destinations.

Some years since we've stayed, on C&CC but looked around this club site. 

Great walking in the area and plenty to explore further afield

There is the Bourton on the Water site which is next door to the train station, but I guess the OP meant a main line station!!

I wonder if the OP found the information provided useful? It would be good to know!