I don't know how useful this will be, but this is my experience of travelling in/cycling with MDF/Plywood - I get my sheets cut by the supplier to A3 size and either hand carry it in by bus or use an A3 backpack art case when I cycle in.
It's tricky, but it is possible to plan around when to bring it in to get it cut, with the small risk that the laser will be in use for a while or the medium risk that the laser cuter is down. Webcams on the laser and tool access (both on the members section of the hackspace site) help mitigate those risks.
Mapac A3 art pack:
https://www.amazon.co.uk/Mapac-Art-Carrier-A3-Color/dp/B003VVY39E/ref=sr_1_9
A3 art folder backpack
I have found that people don't leave quantities of materials in the hackspace as a resource to learn and experiment with. They tend to bring in small samples, for short periods. One could think of it as a project to learn how to use the tool, with a small sample of material to use.