A very quick google search brought up these options
There's also Enterprise & Europcar at Tonbridge.
If you are looking for rental on a Saturday then you will need to get there before midday and they will be closed Sunday
Car hire firms at Tunbridge Wells are based outside the town centre. The station was built 150 years ago before car hire. If you can fix up something, you may need to take a local taxi (which are always available) the couple of miles from the station to the car hire depot.
The above mentioned car hire firms in Tunbridge Wells are both based in Quarry Road, which is at the bottom of Camden Road which is in the centre of Tunbridge Wells, probably a 15 min walk from the station & could take just as long in a taxi depending on traffic volume. From London you would be better of getting out at Tonbridge as the Enterprise office is just around the corner, will probably save you a good hour, has easy access out of the town whereas in TW you will have a hard time finding a quick route out.
I used to live in Camden Road, not far from the Quarry Road end. The first 200 yards is up a steep hill, which I remember less than fondly. I reckon 15 minutes walk also.