My landlords left two vehicles on the property we are renting now in AZ. On Saturday a cover got blown off one and got soaked in a storm now they are beyond upset, and telling my wife and I that we better get ready to move. We have not done any damage to the property itself and even done improvements that they have requested. Since we are late on this months rent, which I told them we would be as soon as I knew, I am afraid they can evict us but they are doing it because of the car and not the rent. So I guess my question is if I get them their rent on the agreed upon date are they able to kick me out over what should be considered an abandoned vehicle?

Do you have a lease? If so when does it expire.

Yes they can evict you, but only by following the laws in AZ. If you are on a lease that has months to go, they can only evict you for a breach of the lease, which would include being late on the rent. They can give you a 5 day pay or quit notice. If there is no breach, then you can fight the eviction at the eviction hearing.

If you are on a month to month basis, they can give you notice 30 days prior to the rental due date, that you need to vacate. There is little defense for that.

They own the land and the house! How can you call their vehicles abandoned?
This story about the cover on one of the cars doesn't sound like it's all told. It sounds like you touched it in some way, and it didn't just blow off on it's own.
Then you say you are late on the rent but that there is 'an agreed on date,' and that doesn't sound quite right either.
It sounds more like the two combined are the last straw and they want you out.