Using a VSAT antenna to track a moving satellite overhead is possible, but it is not the only part of a solution.
You will need to install a lower-frequency receiver and primary radiator in the antenna focus, and use a good mount to allow the antenna to scan the sky. Next you will have to build a "monopulse" primary radiator so that the complex receiver could give you a signal usable to drive the antenna to follow the satellite source in the sky.
One of VSAT antenna feature is that it is an offset parabolic dish that has side lobes directed to the sky. As the LEO satellite moves overhead, your receiver can receive its signal by a side lobe instead of the main lobe, and your system could lose the correct pointing.
Most signal-tracking (radar) systems similar to your project therefore use a symmetrical parabolic dish with low side lobes, and the monopulse primary radiator plus a two- or four-channel receiver, to obtain the pointing drive. If you can sit next to your system and follow the satellite by hand-pointing the antenna, you can follow only the LEO signal level to point to the satellite. Only training will show you how not to get confused by the "high" side lobes of an offset VSAT antenna.
LEO satellites like GPS transmit signals that can be received on the visible(from that satellite) part of the globe, with small not too directive antennas. GPS handheld receivers do not need to follow the GPS satellites, so no tracking is needed. I would advise to test the satellite reception with a dipole or low-directivity antenna; it is possible that the complex tracking system is not necessary at all.