History of forcing over admissible sets

Some of the early results in this erea are due to Jensen. See his notes "Admissible sets", available at his homepage "https://www.mathematik.hu-berlin.de/~raesch/org/jensen.html".

As the introduction indicates, they are based on lectures given at 1969 and section 5 of the paper deals with forcing over admissible sets (and even primitive recursive closed sets).

Forcing over admissible sets was first carried out by Jon Barwise in his 1967 Stanford dissertation.

This recent paper of Mathias (it appeared in Fundamenta Mathamaticae in 2015) develops forcing over set theories that are weaker than KP. It also has a nice historical section (page 31) on the topic of forcing over weak set theories.