Yes in theory, but probably not in practice.
It’s technically possible, yes, but it requires access to edX Studio in order to export the course, and typically only the original authors of the course have that access. Unless you happen to be one of the course authors, you’d have to contact edX or the course author(s) and ask them to provide you with an export.
Since most courses on edX.org are copyright and earn income to support edX and their institution, you would also generally need to work out a licensing deal beforehand, and provide a compelling business case for offering the same content on a competing site.
That will usually be the biggest issue though; copying an entire course is not going to fall under “fair use”.
A few courses on edx.org do use Creative Commons licenses (example), so you can use those ones if you comply with the license terms (in that example, no commercial use is permitted). However, even then, there is no way to directly get an export of the course from the edX UI. So re-using content from a single unit would be relatively easy by copying and pasting, but if you want to re-use the entire course as a whole, you’d still need to ask the course authors to share an export of the course with you or see if they’ve posted a copy of the course export/source code on another website like GitHub.