If a property rental company, essentially a corporate landlord, is looking to acquire homes to rent out on mortgage, would the company take out a personal mortgage or a commercial mortgage, or is there a third type of mortgage just for this situation?
Companies by definition can't take out personal mortgages. They have to take out a commercial mortgage, or other business loan.
A residential property would still be a residential property even if a corporation owns it.
A corporation can not get a personal mortgage or loan but would need to get a business loan to buy the residential property.
If an individual person takes out a loan for residential rental property, they will generally have to meet different requirements regarding down payment. But they will be evaluated based on personal income, personal credit history, and how likely they are to get a renter at the monthly rent need to service the loan.
Once they become a company or they have multiple residential rental properties they will no longer be applying with the part of the bank that handles loans to individuals, they will be dealing with the part of the bank that makes loans to businesses. The will be looking at a different set of numbers: they will be looking at the financials of the business.
Now if the rental property isn't a single residence (single family home, town home, condo) but is either a commercial property, industrial property, or multi family dwelling (an apartment building) they will have have to get a commercial loan. Loans for these types of properties are not made like a regular mortgage to a homeowner.