Chick-Fil-A has generally been one of the worst direct supporters of dominionism on the corporate level as well as via foundations. Examples I've been able to find include stealth evangelism targeting kids--via use of both Veggietales (the creator of Veggietales is linked to one of the early dominionist "charismatic" groups, Maranatha, which was so spiritually abusive it was banned on multiple state campuses) and via giving out Focus on the Family audio programs as children's meal incentives. Chick-Fil-A has also been associated with the promotion of "dad's groups" linked to Campus Crusade for Christ (which has been noted as being coercive on a number of campuses).

Truett Cathy, the head of Chick-Fil-A, is also a major funder of dominionist agendas including National Bible Week (a dominionist campaign in which dominionist groups get state and national legislatures to pass resolutions declaring a week of bible study)--to give an example of just how long dominionist groups have been at work, the first National Bible Week was proclaimed by FDR in 1941. Cathy is one of the most well known promoters of Focus on the Family--a fact that FotF itself promotes on their web pages. Cathy himself also claims to use "Biblical principles" in running Chick-Fil-A and has promoted the Arthur S. DeMoss Foundation--a group that uses sports personalities for stealth evangelism campaigns. Truett Cathy also operates a foundation for funding dominionist causes.