Is it acceptable to use mainly websites in a literature review?
Website or not website is not the point. Printed journal articles are favored over website because i) they are peer-reviewed, carrying some degree of authority, ii) they are archived and retrievable, with payment or free of charge, and iii) once published, the contents do not change until they are formally revised, rebuked, or retracted.

So, it would mean that when citing websites, a few more steps need to be done to increase their worthiness. First, you'd need to assess the credibility of the website and be able to critically evaluate the contents. In journal articles this step is done to some degree but in websites you're on your own. Second, you may need to do most of the archiving (including downloading, dating, and archiving the data sets you may download, or printing and dating website into PDFs, etc.) or use website archiving to save a cross-section of the websites. There are free services like http://perma.cc and http://www.webcitation.org available.

Once they are archived, in your paper, it is a good practice to cite both original link and the archived link as well as the date you accessed the original link. For details, consult the project leader or editor. Some journals specialized in publishing Internet-related research such as Journal of Medical Internet Research may also provide useful format-related examples in their guide to the authors.