How do I reference Lexercise in a research article or grant proposal?

Lexercise is:

1. a Website;
2. a data base that is only available on the Web;
3. a Web-based subscription service.

Therefore, the citation is simply: http://www.lexercise.com/

Following is the detailed  citation information regarding electronic references.

The flow chart in this blog article shows you how to cite a data base that exist only in electronic form:
    http://blog.apastyle.org/apastyle/2009/09/a-doi-and-url-flowchart.html

When citing an entire website, it is sufficient to give the address of the site in just the text.   See:   http://www.apastyle.org/learn/faqs/cite-website.aspx

Example:
Kidspsych is a wonderful interactive website for children (http://www.kidspsych.org).

(adapted from the sixth edition of the APA Publication Manual, © 2010)