Open source software for running Internet psychological experiments that collect reaction time data I've often used Inquisit to run psychological experiments online. The software enables delivery of stimuli (e.g., text, images, etc.) and collection of reaction times. 

Obviously general purpose programming languages provide one avenue for delivery of online experiments. There's also a good listing of software for psychological experiments here. I've also seen PsyToolkit which is GNU licensed software for programming psychological experiments, but I'm not aware of any online option.

Are there any open source options for online delivery of psychological experiments, particularly ones that enable reaction time measurement?
For an open source JavaScript/HTML/CSS solution, check out jsPsych: http://www.jspsych.org. It  can be used for reaction time measurement and interactive designs. An article describing the library was recently published in Behavior Research Methods.

A subsequent article investigated the properties of reaction time distributions collected with JavaScript compared to those collected with MATLAB and Psychtoolbox. The main result is that JavaScript was 10-40ms slower, but had equivalent variance across different experimental conditions and equal sensitivity to the experimental manipulation of set size in a visual search task.

There are several good answers to this related question about the validity and accuracy of response time measurements online. Many of those answers discuss findings that are relevant to JavaScript libraries like jsPsych.

de Leeuw, J. R. (2015). jsPsych: A JavaScript library for creating behavioral experiments in a Web browser. Behavior Research Methods, 47(1), 1-12.

de Leeuw, J. R., &amp; Motz, B. A. (2015). Psychophysics in a Web browser? Comparing response times collected with JavaScript and Psychophysics Toolbox in a visual search task. Behavior Research Methods, advance online publication.