Is There A Website I Can Check To See If The Landlord Is Making The Mortgage Payment?

Unfortunately, I do not think so.  A foreclosure action against your landlord could pose potential problems for you should you decide to rent somewhere else.  A sudden move in the middle of a lease could raise questions from future landlords, so proper documentation by you to support your rental history is critically important.  Some examples of proper documentation to consider are as follows:
*Pay all monthly rent payments and security deposits by personal check, bank check, or money orders.  Paying cash proves nothing in terms of a rental history, and provides no support to you should you seek to recover any security deposits.  Always pay in a format that can prove your honesty.
*Keep copies of cancelled checks to show a consistent payment history. Cancelled checks are the most effective way to show your ability to make monthly rent payments on time as well as the frequency of paying your rent at a certain time every month.  A consistent payment history can provide indisputable proof that you are a good risk.
*If you are served a foreclosure summons, keep a copy for your records. If the rental you are living in does fall into foreclosure, you are likely to be served foreclosure papers as well, since you have a leasehold interest in the property.  If you lose your rental as a result of a foreclosure through no fault of your own, the foreclosure papers you are served along with the documentation noted above will greatly enhance your chances of securing another rental with greater ease, and will also help you tremendously in the event you decide to purchase a home of your own in the near future.
You may not always know if your landlord is making their mortgage payments, but through careful landlord screening and proper documentation, you can protect yourself and minimize the chance of facing eviction by foreclosure.

If you know what bank holds the mortgage, you can check with them. Info only available in person. Take a picture I.D. With that address on it. Any power bills in your name, with that address, to prove you reside there. Rent receipts also. Visiting the bank yourself will also help, if you are forced to move because of non-payment of mortgage.

I would also ask for an updated amortization schedule for your loan.
This way you can track your previous payments.
Also ask for an update on your account that lists
when your payments were made and applied.