Will vault dwellers heal over time?

Your dwellers will recover over time (health and radiation) if you have enough food and water and if they are currently in the vault.

If you lack food, your dwellers will lose health slowly.
If you lack clean water, your dwellers will get irradiated slowly.

Yes, they will recover but it will be very slow and if you have an irradiation higher than 50% or same amount of lost health points, it's not really worth to wait. They will quickly become very unhappy and it's gonna be harder to get back 50, 60 people to 100% happiness (or close). Start creating stampacks and radaways and slowly heal everyone, starting with the most affected ones of course.

What I usually do after something like this, I use a cycle of 1 radaway to each affected dweller (until all the radaway are gone), even though that doesn't completely heal them but stops the rapid happiness decline.