I could only find one linux package at the link you posted - steam_latest.deb
If you unpack and run this, it expects a multi-lib 32-bit/64-bit environment and vulkan drivers - tinycore does not have multi-lib, but it does have vulkan drivers for intel and radeon hardware.
If you use the wine extension to run the windows installer at the link you posted, it brings you to the steam logon - I do not have a steam logon so I stopped there, but things looked to be working (with vulkan drivers).