Since Invision free is a service, (I'm assuming) you don't have access to any of the programming of the board, or the SQL. In order to grab a random post, you would at least need your own hosting and would need to write a particularly complicated script that gathers the text visually.

It would be much easier to setup your own board and then program a random post script as well, but since you're new to webdesign, it would be quite difficult to learn an entire programming language as well as how to set this all up.

Without learning more, I can't see how you can do this!