The Middle Ages is normally said to be between the 5th and 15th centuries (so 400-1400 AD).
Germ Theory was first proposed in 1546 (!), close to the Middle Ages. It's not clear if Girolamo Fracastoro meant cells and bacteria as we know it today, but the core idea is there.
So you really have little to do beyond making influential figures of the period (e.g. A King or two) become convinced it's the right theory and back it. Perhaps another person or your character who proposes the idea is fortunate to have a powerful patron who lets them, e.g. try out improvements in basic hygiene and something as simple as masks and gloves when dealing with blood injuries or similar systematic approaches to dealing with illness. Perhaps the results impress the powerful patron and leads to wider publication and more widespread use. And in time to an early development of the theory and it's applications.
The compound microscope was invented sometime around 1590. Again just outside the Middle Ages. The combination of these two inventions in the right hands and with the right patron would be enough to get things going well. Probably this can be reasonably be made to happen a little earlier (say 1400). The microscope is important as it lets you see something to attach a theory to. "Have a look at this, Oh Wealthy Patron" works a lot better than "I have this wild idea, Oh Great One". It won't prove germ theory, but it will help advance the idea.
A King or Emperor who has a great interest and belief in sciences would be very useful. A King whose heir also has that interest would be better. Science needs money, lots and lots of money. :-)
As an aside you need to explore how science was done in the Middle Ages to get a feel for the way it would happen.