Any valid web application can run on Lampix. The only thing to keep in mind is that Lampix will not, by default, make things interactive.
To integrate with Lampix, use the lampixjs library. The library is meant to be used inside the context of a Lampix device or the Lampix simulator. Loaded inside an environment without the expected API will simply make the library search for the API indefinitely, rendering its most relevant methods unusable.
What web technologies can I use?
You can create your application any way you see fit, provided the end result respects what is outlined in the application structure.
That said, you can use:
VanillaJS - as in no libraries, if you'd prefer not using any
and any other technology used to develop web applications
Do I need to worry about browser support?
You only need to worry about Chromium v66. Search this compatibility table for the functionality you need.
Chromium v66 is quite advanced, and includes features such as the fetch API and web animations API out of the box (the latter, for instance, is an experimental technology whose animation.finished promises show issues, but can easily be mitigated with the properly working animation.onfinish callbacks).