Hi Mark,
Apologies I do not know what APIs are available on the WebEx Connect desktop client. The latest Jabber client (version 9) for Windows supports some protocol handlers to start a chat or a call from a web page, see details here
http://www.cisco.com/en/US/docs/voice_ip_comm/jabber/Windows/9_0_1/b_jabber_win_icg_chapter_01.html#JABW_TP_PF21EDAF_00Regarding the WebEx Meeting Center, I think using the meeting URL as the "href" attribute of the link in your web page, along with "target=_blank" (optional) should open a new tab or window and load the meeting center. Similar to clicking a meeting invite in an email.
Merry Christmas ! Thomas