We are moving out of an apartment in Cheney Washington and our landlord is charging us a fee for having pets, but we have never had a pet of any kind in the apartment. Is that legal?
The relevant law is RCW Ch. 59.18. The first thing is, a landlord can't charge anything that is not specified in the lease. The landlord can require a pet damage deposit, in case the pet does provable damage. Starting with §260 you can read about deposits – I assume that the landlord is refusing to refund the full deposit. It is allowed to hold back on the security for tenant non-performance (failing to vacate on time or pay the rent).
If the landlord does not refund the deposit as legally required, you can sue them in small claims court for intentionally and wrongfully not refunding your deposit, and you can be awarded up to twice the deposit. If the lease states that there shall be e.g. a $25 monthly pet fee and the landlord claims to have "discovered" that you had a pet for a year, they can deduct that amount from the security deposit. In court, you can dispute the allegation, and the landlord will be required to prove both that there is such a provision in the lease, and that you had a pet.