I wonder if there are internet platforms that allow recording the response-time to online tasks.
Yes. Qualtrics, for example, allows this, and has a free trial (or did, anyway). Also, any web survey software (installed or SaaS) can do this providing you can customize your tasks using javascript, java, flash, etc. Most languages/environments designed for programming scripts or interactive media have built-in functions, objects, etc., like "button.press" or something that allows you to ensure something is done when a button is pressed. This includes recording a time, such as the start time if the button pressed begins a new task or the button pressed is the response (often, the same button may both record an end time and start time).
More importantly, there exist pre-built libraries and scripts so that you don't have to write your own code. For example: www.jsPsych.org/
and see
jsPsych: A JavaScript library for creating behavioral experiments in a Web browser
(https://www.academia.edu/attachments/33355670/download_file?st=MTM5ODk4OTE2MSw3MS4xMjcuNTUuNjUsNDMzNzg2OQ%3D%3D&ct=MTM5ODk4OTIwOCw0MzM3ODY5)
[Full citation: de Leeuw, J. R. (2014). jsPsych: A JavaScript library for creating behavioral experiments in a Web browser. Behavior research methods, 1-12.]
The one thing you have to worry about is how precise you wish to be. Behavioral experiments that require very accurate RTs are not suitable for web/online studies because there is too great a delay between the time the participant responds and the time the script, applet, etc., records that time. However, for other purposes there's no problem. See e.g.,:
Dandurand, F., Shultz, T. R., & Onishi, K. H. (2008). Comparing online and lab methods in a problem-solving experiment. Behavior research methods, 40(2), 428-434.
(http://www.psych.mcgill.ca/perpg/fac/shultz/personal/Recent_Publications_files/online08.pdf)
Keller, F., Gunasekharan, S., Mayo, N., & Corley, M. (2009). Timing accuracy of web experiments: A case study using the WebExp software package. Behavior Research Methods, 41(1), 1-12.
(http://www.researchgate.net/publication/23963998_Timing_accuracy_of_Web_experiments_a_case_study_using_the_WebExp_software_package/file/d912f510274ace758d.pdf)
Reimers, S., & Stewart, N. (2007). Adobe Flash as a medium for online experimentation: A test of reaction time measurement capabilities. Behavior Research Methods, 39(3), 365-370.
(http://wrap.warwick.ac.uk/613/1/WRAP_Stewart_Adobe_online.pdf)
Article Timing accuracy of Web experiments: A case study using the W...
Dana, if you are really worried about reaction times, you can try to triangulate your results with eye tracking. For example, what is the time difference between gaze "finding the solution" and performing/recording the related actions through the UI.
Since last week, it is also possible to use Qualtrics + a few javascript snippets in combination with our QRTEngine. See http://www.qrtengine.com/
Barnhoorn, J., Haasnoot, E., Bocanegra, B.R., & van Steenbergen, H. (in press). QRTEngine: An easy solution for running online reaction time experiments using Qualtrics. Behavior Research Methods.
http://www.leidenuniv.nl/personal/steenbergenhvan1/docs/papers/Barnhoorn%20et%20al.%20(2014)%20QRTEngine%20An%20Easy%20Solution%20for%20Running%20Online%20Reaction%20Time%20Experiments%20Using%20Qualtrics.pdf
Article QRTEngine: An easy solution for running online reaction time...
Disclaimer: I work for Psychstudio. I'd highly recommend https://www.psychstudio.com for web based reaction time experiments as it has highly accurate timing and high precision stimulus event coordination. If you do not want to code then do give Psychstudio a try (it is completely UI driven). You can sign up for a trial account and create a few experiments to try it out.