Is it possible for your Companions in Fallout 1 to wear better Armor or are they stuck with what they start with?
I tried giving better Armor to Ian & Tyco but it seems to just sit in their Inventories.

There is a specific mod for that. TeamX NPC, to be exact. Check the Fallout 1 mods sticky.

No way to do it in the "Vanilla" version, huh?

I'm well into a playthrough...I am not great with Modding, but adding this Mod would probably "break" my current game wouldn't it?

No, its not possible in "vanilla" Fallout1. You must use either separate mod for this called NPC MOD or multi-mod pack like "FIXT". I can't recommend you FIXT, it made me angry after several problems with stability and game hanging when saving. I plan to try with separate NPC mod/
You woould have to start new game in order to use this mod and allow your companions use armors.

good luck :)

During one of my play throughs I installed a mod for this and it didn't give me any problems. It was kind of fun, something new. Plus it looks cool once the entire party has combat armor. I remember that I didn't want to wear power armor just so I didn't spoil the image :D On the other hand, equipping everyone with powerful armor sets makes you quite overpowered. Another mod boosting combat difficulty would be quite nice.