I'm playing a level 1 artificer and my GM will have firearms inside his campaign, so he let me take the Firearms Proficiency. My question is, if I get my starting equipment based on the artificer starting equipment page, one being a Light Crossbow, does the Firearms Proficiency work with the Crossbow I have or would I have to wait for an actual gun?

You'd have to wait for an actual gun. Crossbows are, depending on size, either simple or martial weapons.

Generally speaking, I'm wary of any homebrew rules that codify ways to gain more than tool or language proficiencies through downtime training. Unless your campaign is structured in such a way as to put pressure on the PCs' time and/or money, it incentivizes players to just sit in the safety of town to train and check off more boxes on their character sheets, which... isn't particularly fun or engaging.

So what's the actual problem that you're trying to solve? Do you want to give out this cool firearm as a reward, but are worried that it's basically useless if the PC isn't proficient? You can just rule that the firearm counts as a martial weapon. Do you specifically want to include the narrative beat of having the PC learning to use a firearm from an instructor? I'd personally just DM fiat grant the PC proficiency through a special boon/questreward, rather than fiddling with am overarching homebrew training system https://100001****/ https://1921681254.mx/  .

Artificers automatically get proficiency with firearms as per their rules.l, provided that firearms exist in the DM’s campaign. No, it won’t count towards crossbows, but artificers also have proficiency with all simple weapons, and the light crossbow is a simple weapon.