@Jon O'Callaghan
Hi Jon
Can I ask what kind of information you are looking to store for your students? Is it quiz scores, answers to assignment questions? Or is it just what modules they have completed?
I, also, am not familiar with Google Classroom but most LMS's will track module completion. There are a number of ways of storing other types of information. Many do involve extra costs but there are also some things you can achieve which are free or involve minimal costs. It depends what you need.
The main functions of an LMS are:
(a) 'bookmarking' - allowing the learner to pick up a course or module where they left off
(b) completion tracking - signalling to the learner (and the tutor) which courses/lessons/modules have been completed
(c) Recording test scores - storing the pass/fail status and or quiz or assignment scores of learners for each module
Still the most common standard used for passing information from the eLearning to the LMS is SCORM - even though this is a very old (and seriously limited) protocol.
Increasingly, institutions are using the more modern xAPI protocol which was designed as a replacement for SCORM. xAPI allows the developer to define pretty much any kind of information they want to store and then add code that will send that information to the LMS - or, more accurately, to an LRS (Learning Record Store). Many LMS's now have an in-built LRS to accommodate xAPI - BUT it is also possible to purchase access to a separate LRS.
It sounds as though the best route to meet your needs would be to explore the less expensive LRS options (there is only one open source "FREE" one I know of - but it's a WordPress plug-in)
OTHER Options
You could add JavaScript code to your Storyline modules that would store information in Google Docs (a Google sheet, for example) - or in a Microsoft Teams spreadsheet (Excel).
You could add JavaScript to your Storyline modules that added learner input to a pdf file that the learner could download at the end of a module.
...I'm sure there are more options others can suggest. Sorry this answer was a bit long - hope it helps?
Regards