“When you get bit you turn into a zombie”
Simple. Germs/mites etc. that reproduce in zombies and cause the condition also congregate in the mouth. Lots of real viruses do this.
While simple, this is not a necessary feature of classic (Romero) zombies. Arguably, they spread their condition simply by inducing fatal sepsis from a mass of mundane germs. Their condition is death itself while under the influence of some unexplained cosmic force, i.e. magic.
“The body is dead but still able to move”
It would have to be moved like a puppet. This would require the cause of zombification to digest the rotting human musculature, breathing apparatus etc. and to replace these with new and effective organs. This is unheard of in nature, and not practical for a bio-weapon.
I would argue that the body does not have to be dead. Settle for brain death or a debilitating change in temperament à la 28 Days Later (2002). The mites/germs rebuild parts of the brain to control the body, causing typical zombie hygiene as a side effect.
“Zombies can only be killed with a head shot”
If head shots are highly effective, that would be explained by the mites/germs using the brain in the manner of cordyceps fungi.
If all other damage is less effective than it would be against a living human, that would be explained by the destruction of relevant nerves in the zombification process to eliminate pain, coupled with enough endocrinological mayhem to push the body to brief peaks of hyperactivity in search of food, new hosts etc., but a shot through the heart would still stop a zombie.
A less credible disease vector could conceivably combat infections and other mundane causes of death to keep its host mobile, and clot blood better than we do, and so on. A still-less credible vector could repair its host. An incredibly fancy bio-weapon could build a redundant heart.