You can fire an event from the field that the value picker updates.
Here's a simple example that updates another field when the value picker is used:
<xe:djextListTextBox id="inputField">
<xp:eventHandler event="onChange" submit="true" refreshMode="complete">
<xe:this.action><![CDATA[#{javascript:getComponent("testField").setValue(getComponent("inputField").getValue())}]]></xe:this.action>
</xp:eventHandler>
</xe:djextListTextBox>
<xe:valuePicker id="valuePicker1" for="inputField">
<xe:this.dataProvider>
<xe:simpleValuePicker valueList="1,2,3" valueListSeparator=","></xe:simpleValuePicker>
</xe:this.dataProvider>
</xe:valuePicker>
<xp:br />
<xp:inputText id="testField"></xp:inputText>