Answer
Yes. You can automatically connect to ArcWeb Services each time you open ArcMap by setting to true the RECONNECT_AT_ARCMAP_START value in your login.xml file. To change this value, open your login.xml file in a text editor and set the RECONNECT_AT_ARCMAP_START value to True.
<RECONNECT_AT_ARCMAP_START>True</RECONNECT_AT_ARCMAP_START>