What is the history of self-experimentation in cognitive science? I am looking for a source on the history of self-experimentation in the cognitive sciences, or experimental psychology in general. I am specifically interested in knowing when it stopped being the norm.

The Wikipedia article on self-experimentation has no sources for psychology.
Short answer
Based on the general developments in drug and medical device development research, 1970 was a major milestone, as randomized controlled trials were marked as compulsory for new drugs to be marketed through the FDA regulations. 

Background
The question is quite broad, but in terms of clinical trials I can say that since 1970 the FDA required that drug producers submit randomized controlled trials (RCTs) results with new drug applications, heralding a new era of the RCT in drug approval regulations (Meldrum (2000); Bothwell &amp; Podolsky (2016)). An RCT is defined as 


  [A] type of scientific (often medical) experiment which aims to reduce bias when testing a new treatment. The people participating in the trial are randomly allocated to either the group receiving the treatment under investigation or to a group receiving standard treatment (or placebo treatment) as the control. Randomization minimizes selection bias [...]. The RCT is often considered the gold standard for a clinical trial.


Now you are referring to questions on Cognitive Sciences or Psychology in general. Drug research and medical device investigations include the widely used psychoneuropharmaceuticals such as the antidepressants and antipsychotics, as well as medical devices such as deep brain stimulators and sensorineural implants to treat e.g.., epilepsy and deafness/blindness, respectively. All of these can be included in the disciplines of Cognitive Sciences and Psychology. 

References
- Bothwell &amp; Podolsky, N Engl J Med (2016); 375: 501-04
- Meldrum, Hematology/Oncology Clinics of North America (2000);
14(4): 745-60