The quick answer is "no". Your client can not "tap" into someone else's MLS data.
If your client has a user name and password for their own account on MLS, usually what we program for is a download of their properties overnight and then write code to place the data wherever on a Web page. Typically, after the initial download (many, many megs) you only download the updates with a weekly complete download.
If you would like to pursue this further, please feel free to contact me at XXXXXXXXXXX
Thank you,
Cassie Woolworth