At first, the earliest reference I could find to a spring seat in a vehicle was in William Felton's A Treatise on Carriages (published 1794), in which he describes (on page 144):
it [the coach-box] is fixed on to the top iron-work with bolts, having a cradle, the same as the others, for the seat; they sometimes hang upon springs, and are made with a head and knee-flap the same as to a one-horse chaise; their use is to make the situation of the servant more comfortable, and more secure from danger, by travelling on bad roads
This source seems to take the existence of the spring seat for granted, so it seems to already be implemented at this time, meaning the spring seat may have been around for longer. However, the Gold State Coach (commissioned in 1760) does not seem to have a sprung seat, so the innovation probably occurred at some point between 1760 and 1794.
Finally, I found an early patent that coins the word "spring seat". It is dated "A.D. 1792, May 25" by John March.
Certain springs and rollers are set forth as being placed below the body of the carriage, apparently for the purpose of preventing such body from swinging improperly from side to side owing to its "centrical suspension," and both the coachman's seat and the footman's standing board are also mounted on springs, there being also a "spring seat" for the inside of the vehicle which is supported by hooks and straps, as well as other seats or stools inside the vehicle which turn upon hinges so as to be raised or lowered at pleasure, and likewise a "drop table."
This patent is described in Patents for Inventions. Abridgements of Specifications Relating to Carriages and Other Vehicles for Common Roads. A.D. 1625-1866. London.