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).
I only have intel hardware to test.
Xorg-7.7-3d contains /usr/local/lib/libvulkan_radeon.so
If you'd like to test, you can use vkcube and vulkaninfo from the vulkan-tools extension.
tc@box:~$ vkcube
vkcube: /usr/src/Vulkan-Tools-1.1.96/cube/cube.c:3148: demo_init_vk: Assertion `!err' failed.
Aborted
tc@box:~$ vkcube-wayland
sh: vkcube-wayland: not found
Did you install Xorg-7.7-3d, graphics-KERNEL, the appropriate xf86-video-* extension and the appropriate graphics firmware extension?
$ tce-load -wi graphics-4.19.10-tinycore alsa-config Xorg-7.7 xf86-video-ati firmware-radeon flwm wbar aterm alsa pavucontrol
$ sudo /usr/local/etc/init.d/dbus start
$ startx
Steam.tcz >> https://store.steampowered.com <<
https://store.steampowered.com/search/?os=linux
You need Xorg-7.7-3d, but you loaded Xorg-7.7
$ tce-load -wi graphics-4.19.10-tinycore alsa-config Xorg-7.7 xf86-video-ati firmware-radeon flwm wbar aterm alsa pavucontrol
As stated previously, there are no plans to make a multi-arch version of tinycore.
At the moment steam needs multi-arch so it will not work.
$ tce-load -i graphics-4.19.10-tinycore alsa-config Xorg-7.7-3d xf86-video-ati firmware-radeon flwm wbar aterm alsa pavucontrol[\code]
As stated previously, there are no plans to make a multi-arch version of tinycore.
At the moment steam needs multi-arch so it will not work.