How many missions would it take to make lave Federation?
It doesn't work like that, I'm afraid.
Lave has a population in the billions, iirc, and is a well-frequented hub due to its rare good - Lavian Brandy - and its resonance in the history of the game.
Trading at a station and buying goods from it (Lave Station is also an agricultural station, so it attracts a lot of traffic from CMDRs running missions for other systems) buffs the influence of the faction in control of the station. Population acts as a negative modifier against the impact of player actions.
Without knowing the exact algorithms involved in calculating influence changes, the number of CMDRs through the system, and every one of their actions trading and shooting, etc., it's impossible to give you any firm idea of the number of missions it would take to flip the system. I'm pretty confident in the answer "more than you have time to do on your own." Additionally, the size of the population means that it would take an organised and well-coordinated effort from a group of CMDRs to have any noticeable impact at all - there's a cap on influence swing which is based on the amount of traffic and the number of actions performed by CMDRs in the system.
One dude on their own isn't going to do squat. I'd add that if you tried to flip Lave to the Federation (does it even have Fed-aligned factions?) you'd stir up a hornet's nest of very angry Alliance-aligned players who would very quickly put a stop to that kind of thing and write strongly-worded letters to their local political representative before running hundreds of missions and utterly negating your effort.