Are agency carers allowed to give care in a home that has a rat problem? I found a dead rat in the lady's kitchen cupboard.

You should notify your supervisor and the county health department if you think the premises is unsafe.

If your agency continues to send caregivers to this lady's home the agency needs to get in there and do a status check on your client so the agency can see for itself what kind of conditions in which this lady is living. Until then, if you're uncomfortable, refuse to work in this lady's home. It's your agency's responsibility to contact authorities if they see fit to do so. It's your responsibility to not work in an environment in which you're not comfortable.

I'm assuming you have reported to your agency and nothing has happened/ changed?
I would contact your state's department of elder care and report it to them.