I can't see any sign of Army records for William RENDER, in the RHA,
only a Corporal William RENDER in the Waterloo Medal Roll, but he is in the 12th (or PRINCE OF WALES) REGIMENT OF LIGHT DRAGOONS
The only ofther information I can find is the following.
MARRIAGE
Tynemouth
William RENDER & Jane NICHOLS 21 Jan. 1807
BURIAL
St. Andrew Newcastle
Geo RENDER Darnells Buildings St Johns, 28 Feb 1809 age 6m son of Wm (private Royal Artillery)
BAPTISMS
St. Andrew Newcastle
Thomas RENDER 16 Oct. 1814, parents William & Jane RENDER
Jane RENDER 18 Nov. 1816, parents William & Jane RENDER
St. Nicholas Newcastle
James RENDER 27 Aug. 1826, mother Jane RENDER
From this it would seem that James was not William's son, as he was born seven years after William died, as is born out by the ages he gives in the census.
Jebber