When market equilibrium occurs, quantity demanded is equal to quantity supplied, which means that both sellers and buyers get what they want. Does a market reach market equilibrium on its own, or is it necessary to have some sort of regulator to manage the price and ensure there is equilibrium?

Whenever you have a buyer and seller, the two sides will come to some agreement as to what to pay. The price might start off too high or too low, but eventually it settles at the equilibrium. It's just the nature of markets.

So the market reaches equilibrium on its own. It doesn't need any external forces like a government to come in and force a certain price.

If anything, a government usually moves the price away from equilibrium. Example: if gas prices are too high, then the government could set a price ceiling which would lower prices but would lead to gas shortages like in the 1970s.