You cannot dispel Zombies in D&D 5e
Animate Dead, the spell that creates zombies is a spell with an instantaneous duration. You cannot dispel effects of spells with an instantaneous duration.
This has been explicitly answered in the Sage Advice Compendium:
Can you use dispel magic on the creations of a spell like animate dead or affect those creations with antimagic field?
Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend. Here’s why: the effects of an instantaneous spell are brought into being by magic, but the effects aren’t sustained by magic (see PH, 203). The magic flares for a split second and then vanishes. For example, the instantaneous spell animate dead harnesses magical energy to turn a corpse or a pile of bones into an undead creature. That necromantic magic is present for an instant and is then gone. The resulting undead now exists without the magic’s help. Casting dispel magic on the creature can’t end its mockery of life, and the undead can wander into an antimagic field with no adverse effect. [...]
Of course there are many other ways to off zombies, from just beating them down, to a cleric of level 5 or higher using their Channel Divinity to destroy them. (Thanks to @KorvinStarmast.)
Wether you can reuse the corpses of zombies that were destroyed in some way in 5e is covered by Can a Necromancer reuse the corpses left behind from slain undead? in detail. Short answer (also based on the SAC): yes, if the creature at some point was a humanoid.
Can I cast animate dead on the humanoid-shaped corpse of an undead creature such as a zombie or a ghast?
When animate dead targets a corpse, the body must have belonged to a creature of the humanoid creature type. If the spell targets a pile of bones, there is no creature type restriction; the bones become a skeleton.