Bees can only survive about 3 weeks without a queen. If they are unsuccessful in raising a new queen on the first try, they will be in very severe danger of dying. After about 3 weeks without a queen a hive will begin producing laying workers See: Laying Worker. Laying workers can only produce drones- which are useless to the hive.
Hopefully, when you discover your hive is queenless it will have been less than two weeks.
What to do…
There are several different clues that will tell you how long the hive has been without a queen. If there is still some young brood it will have been less than a week and a half.
In this case, if you can simply give the hive a new queen you will (probably) be fine. The problem arises when you can not find a supplier for a queen. Don’t despair though! you still have a few more options. You can let them raise their own queen. This has about an 80-90% success rate. However, your hive’s growth will be severely stunted. If you are able, give them some frames of brood. A hive loses roughly 1% of its population every day. If there is no new brood emerging the population can decline very quickly.
Another option is to combine the failing hive in with another hive. You still lose a hive, but all those bees live (so it is not a complete loss). Who knows, you might be able to split the combined hives later in the year if you can get access to some queens!
If you are able to get ahold of a queen, or you decide to let them raise their own, it is also very helpful to give them some fresh brood covered in bees. This will help hold them over until the new queen can start laying.