It's almost correct. Change to "A dog that bites someone..." or "Dogs that bite people..." to make the sentence correct. Using "that" is fine in this sentence.

>Dogs that bite people must be quarantined for a period of 2 weeks unless proofs of their recent rabies vaccinations are provided / Dogs that have bitten someone .. Proof is more certain .Its either right or wrong for the reason . Evidence can be weak or false or disputable. "That" precedes a restrictive clause, and in this case , it's " bite people" .So, we are talking about these dogs and not others . "Which" precedes a non-restrictive clause , and the clause must have commas before and after it . It is used to give additional information for the subject without spoiling the intergrity of the sentence when it's ommited >These dogs , which are vaccinated against rabies , will not be quarantined. Something "which" can br restrictive , and interchangeable with "that "> The cat which/that is lying on the floor over there is mine . But, for your case , I would stick with their general usage , and that is "that"