I think you'll struggle with those three locations. When I lived there, there was absolutely nothing, and it's even more built up now. If you can stretch to Sanderstead, New Addington or Warlingham you'll have much more choice and much better riding to boot.
ps If you're wanting to stick on the tram route, you might be as well looking at Wimbledon too (although then you're talking expensive)
There used to be yard near to Lodge Lane going from Addiscombe towards Addington. I think it was called Haethfields Farm or something like that. Other than that you'd be struggling as not a very rural area. Keston isn't far away though and as stated above Warlingham is pretty horsy
Thank you guys Esther, I had a feeling this area was not so good for options, New Addington could be ok, any places close to the tramlink there that you know of ?
Like you say, Wimbledon is pricey but so easy to get to...
Unfortunately I haven't lived down that way for years, so I can't help - a lot of the places I used to know are now houses or golf courses. If heathfields is still there, that would be close-ish to the tram (near Shirley Hills). When I've been down to visit my mother I have seen the odd horse riding out so there must be at least one yard locally still.
Church Farm is right next to Woldingham Station (which is on East Croydon train line). Heathfields is still there but large mixed herd. Few in Keston but without car youd be walking down dark country lanes. Wickham Farm in West Wickham about 10 mins from bus stop (or cut across field 5 mins).
Im in area so feel free to pm me as have probably looked at everything in a 10 mile radius we we moved.