Hello Catrine,
This is a very late answer to your question, but since folks might still be searching for a solution, I figured I'd reply.
There is a product called LabMaestro Pack&Go by VPixx Technologies, which allows you to run MATLAB experiments online. You can upload your studies to our remote servers and stream the experiment to your participant's computer. All your participant needs is a stable internet connection.
While it is not integrated natively with MTurk, we support Prolific (another recruitment platform).
Sincerely,
The Team at VPixx Technologies
