You will need to customize the QUERYFORM in ArcPad Studio to add a control which references the ObjectID field.
1. Open your AXF in ArcPad Studio.
2. Under 'Feature Layers' double-click on the feature class for which you want to search ObjectID's.
3. Under 'Forms' double-click on <QUERYFORM>.
4. On an existing page (or a new page that you add) drag over an Edit Control from the Controls Palette. The Edit Control has an "ab" on it and is the upper right-hand corner of the palette.
5. Double-click the new control that you've dropped onto your edit form. Find the 'Field' dropdown menu and set it to OBJECTID.
6. Optionally, drag a Label Control (top-left of control palette) so that you can label your new ObjectID control.
Now when you open the binoculars in ArcPad, you will be able to enter values in this new control and it will search on the ObjectID.
To use this form for later checkouts, while on the first page of the Get Data For ArcPad wizard, specify the action for checking out your feature class and then click the small check box to the right of the feature class. Navigate to the AXF file you altered in Studio and it will pull the customizations from it.