Buy a mated queen. Letting them make their own is 24 days minimum to a laying queen, and there are no guarantees. You do not have that time available. The season is short. Your window for that opportunity has passed both in terms of the days left in the season and in terms of the biology of the beehive with and without a queen. By the time they get a new queen going from donor brood, all of the existing bees will be dead or nearly dead. The population will crash spectacularly. The new queen will be the last one standing.
The player is on skip, repeat, ... if you want to still have a hive, buy a queen. If you want to watch, for educational purposes, how a hive crashes and dies out go ahead and wait for them to try to make one.
PS1: Any eggs you see the hive make now is very likely going to be laying workers, unless you lay eyes on a big fat long mated queen.
PS2: Your location is in Montana. I guarantee if you do not have a laying queen in 7 to 10 days from now the hive will not survive the coming winter.
PS3: I suspect this hive is already done for, finished. It has been too long without a queen. Do you have other hives or just the one. If just the one, perhaps you will be best off to just let it runs its course and die out. You will get more honey. Start planning for next year. Next spring install a new package and try again.