Do I get a weapon in Animal Hospital Anomaly?

Yes, you can fight back against monsters, but most weapons unlock over time:

Taser: A free taser appears in the Check-In Office after Shift 4. Tasing a fake patient rewards +2 sanity, but tasing a real patient costs you -15 sanity, so only use it when you are sure.

X-Taser: The Security class starts with an X-Taser (5 uses), letting you handle anomalies from Shift 1.

Gun: The Secret Agent class starts with a gun (20 uses) whose ammo regenerates over time. It is a premium Robux class.

Bare Hands: If you have no weapon, you can punch an anomaly off a patient by spamming E (or hold E from behind if it is attacking).

See the class tier list for which classes start with weapons.