I believe nohillside and fsb are indeed correct.
Monitoring the status of a website is something that iOS was never intended to do, except manually. By actually visiting the website and seeing if it is up or not. Using a phone to do such tasks (assuming it is even possible) will likely eat up battery power and might even fail depending on how strong your local WiFi or cellular signal is. Remember you are running a background task that is always running in the background. A service that needs to use the WiFi/cellular radio. All of which have a cost in battery life.
So it is likely your best bet is an internet service that monitors websites. On my Wordpress blog there is a feature built in to one of the standard plugins to notify me if it goes down. There are other services like Pingdom (no doubt you can find a free one) that do exactly what you want and will notify you of downtime.