Soon to be homeless (Hinckley)

We are a family of 4, about to be homeless.
We've looked for apartments in our area and haven't found anything.
A house to rent would be more ideal than an apartment, my son is autistic and screaming/being loud is his stim.
We also have pets that we consider family, they're very friendly and good in the home.
We can pay up to $1200/month, with first, last and deposit ready.
Any and all help is welcome, thank you in advance.