If one of my suppliers has a system outage, does my whole booking website stop working?

No, your website should keep working, as long as you have more than one supplier connected for that type of product. Each supplier sits behind its own connection, and a search asks all of them at the same time rather than one after another. If one stops answering, the search waits a short while, gives up on that one, and shows the customer what the other suppliers sent back. The page still loads and bookings can still be made.

What you do lose is that supplier's content while they are down. If they were the only source you had for hotels in a particular country, or the only airline content for a certain route, those results simply will not appear until the supplier comes back. Customers do not see an error, they just see fewer options, which is why it is worth knowing which parts of your inventory depend on a single supplier.

Bookings that are already made are a separate matter. A change, a cancellation or a voucher reprint has to go back to the same supplier that holds the booking, so while their system is down those actions may have to wait, or be done by phone or email with the supplier directly and recorded in your system afterwards. How much of this can be queued and retried automatically depends on the supplier and on what their API allows.

It is worth asking each of your suppliers about their planned maintenance windows and whether they notify you of outages, and asking your technology provider whether a slow supplier is skipped automatically after a set number of seconds and whether someone on your side is alerted when a connection fails.