||Yes you can; the information on how to is detailed on page 46 (GPS Height) of the ArcPad Reference Guide.
You need (1) a GPS that has accurate height data.
(2) The Geoid / Ellipsoid (WGS84/Local Elevation Datum) separation at your work location, which you enter into the GPS Preferences. (3) the offset from the ground to the phase centre of your GPS Antenna which again you enter into the GPS Preferences.
For this to work with the NMEA protocol, the GPS must report the separation it is using in the NMEA output sentences.
It Does work, I am using it. If you have a .bin format Geoid/Ellipsoid grid file I can give you the code to compute the separation at the GPS location.
Trimble Receivers using TSIP are more directly supported.