It seems that their response is what you need to do: buy one of the APIs they mention. You can check what you got with the system in case you already have this software, but if not, you will need to buy either API-1 or API-2 if you really want to interface this to LabVIEW. As for which one you need, at a cursory glance it would seem that API-2 is the better choice, but you really need to talk to the manufacturer and tell them exactly what you want to be able to do, and they can tell you which one is more appropriate.
I should warn you: this clearly requires interfacing to a DLL, and this is not trivial. If you lack programming experience and more specifically, LabVIEW experience, this is not something you will likely be able to do on your own as a first project. You should consider whether it's worth trying to interface this to LabVIEW, or just sticking with what they've got and coming up with an alternate method.
Just my $0.02. This is not meant as a discouragement, but as a reality check.