Thanks for your question!
Currently internet is available in Cape York though Skymuster NBN, which utlises fixed satellites to deliver NBN connection. You can find more information on this website here.
In terms of your question about how internet is accessed, I am unaware of any specific data regarding this, however through a google search I found this Remote Indigenous Communications Review report. On page 87, it gives specific information about both the current infrastructure in Cape York, connectivity in certain areas, as well as gaps that could be addressed. The report also outlines general information about projects in place to increase connectivity which you might also find useful.
Some other points for your research could also be taking a look at the Census data for Cape York, which has a section on internet usage, and this document (found in the ICT Design Area resources) which gives some tips about how to understand mobile coverage. You can also see an example of current internet infrastructure in our Interactive: Explore CfAT project examples (found in Resources) in the walkthrough of a Ranger Base.
I hope you find these resources useful in framing the current context, please reach out if you have any further questions!