I have to confess that I cheated a bit and bought spores from the catalogue after using the BB.showhidden objects cheat. Then, after wasting loads of vaccine on sims who would not help me fight, I ended up buying vaccines too. I would have gone mad if I'd had to do it the hard way.
At the time, I did not realise that you can only have a maximum of three sims to help you fight, so I wasted time and resources trying to build up a bigger team. Basically, all you need are three sims with whom your sim has a friendly relationship of 50% or more, vaccinate them, recruit them and take your team straight to the lab.
Incidentally, something else I only found out the hard way, is that when you fight, you are supposed to direct how the fighting is done. I don't want to give too much away here, but don't just start the fight and hope for the best - that way you are likely to lose. My sims fought and lost four times, before I realised there was an option to direct their actions. Then they won easily.