You will need to set the text adapter's start method in either a monitor all state to monitor whenever the Host On Demand application starts on the desktop by user interaction, or create an additional web adapter in which you would automate the necessary steps to launch the host on demand through internet explorer.
Below is also a link to an article that should help you with setting up the host on demand text adapter as well.
The installation of the hllapi bridge component is something that must be done by the IT team. The IBM Host emulator has to be set to use the Ehllapi or WinHllapi dlls. It doesn't sound like the Emulator is currently setup to communicate to the mainframe through hllapi function calls.
Please have the IT team get in contact with the vendor on configuring the Host emulator to being hllapi 32 compliant. Once that has been done, you should be able to integrate with the emulator using the steps outlined in the article provided.