As I understand it, we don't make prayers of request (bakashot) on Shabbat (cf Rambam Shabbat 30:12). For that reason, we omit so many liturgical requests because of shabbos - e.g., Avinu Malkeinu, tachanun, the 13 attributes, singing during birkat kohanim, most of the amidah.
If that's so, then why are we allowed to make requests during the Torah service - e.g., mishebeirach, birkat hachodesh, yekum purkan, Kel malei rachamim, prayer for the country, prayer for Israel?
Taking out the Torah constitutes an eit ratzon, a time appropriate for making requests to the King Whose presence we have invoked.
Here's another article about it which elaborates on the concept of eit ratzon and why this supercedes the general issue of not making requests on Shabbat: http://www.beureihatefila.com/files/2010-06-04_Tefila_Newsletter.pdf
The whole point is that the eit ratzon appropriate for making requests on Shabbat depends on taking out the sifrei Torah.
