Hi,
I think you can do. I couldn't test it now on field, but it seems that works.
Obviously, you need two layers selected for editing in ArcPad - 1 point and 1 polyline.
When you capture polyline using GPS and you "run into" a feature you have to log as a point, simply just select the Point tool from the Point drop-down menu and capture the point (select Capture point w/ GPS too). When you finished, just switch back to polyline tool (and select Capture Vertex w/ GPS as well) to continue capturing the fence or whatever you need...
(Same method works with manual capturing too.)
Test this process before your client goes to field, as I couldn't - I don't have any GPS to connect to ArcPad at the moment - but it should work.
Hope this helps.