Option A - Download firefox itself.
You can do this by searching up "Firefox" on your current browser, find the official website, and click the download button. Then, transfer your steam passwords and others to the firefox site, and you can freely search any mod on the half life 2 workshop. I have tested this, and so far this option has been working.

Option B (Not recommended as it is less good for mod hunting) - Youtube.
Simply by using youtube and finding half life 2 mods with the mod links, you have a high likelihood of accessing the mod without downloading firefox. This is not recommended because it lacks the ability for you to search your mods.

These are the two only options. If you have developed another way of accessing the HL2 workshop, reply to me, and I will likely add it.