[puréé]

If I drop a book here on Earth, flat earthers would say it fell because it was heavy. That leaves me with one question: why doesn't the moon fall to the Earth? 
Now disprove that Earth doesn't have some sort of gravitational pull.

[Bastian Baasch]

If I drop a book here on Earth, flat earthers would say it fell because it was heavy.

Wrong, flat earthers would actually say the earth accelerated upward and caught up with the book.

That leaves me with one question: why doesn't the moon fall to the Earth? 
Now disprove that Earth doesn't have some sort of gravitational pull.

I think you can be forgiven for this one as it's not in the FAQ, but flat earthers do believe the earth has a gravitational pull, they call it celestial gravitation.
https://wiki.tfes.org/Celestial_Gravitation

Basically, it's a stopgap to a hole in their model, as one can see not only from the shortness of the wiki article, but the fact that selective gravitation doesn't really jive with flat earth, they have yet to answer why it works on such a large scale, what property is responsible for it, if it is responsible for the tides, a moon to ocean interaction, then why are there no mass to mass interactions on earth?

[manicminer]

Like all orbiting bodies the Moon is in a state of constant free fall. That means technically it is falling towards Earth. The angular momentum (constant state of acceleration) prevents it from falling into the Earth. The Moons distance from Earth varies during its orbital period. At its furthest point (apogee) it moves slowest while at its nearest point (perigee) it moves fastest.

If the Moon was to have its orbital velocity altered for any reason then its distance from us would either increase of decrease accordingly.