Can the previous owner rent from me if I buy their house on a short sale?
Once you are the owner of a home, you can do whatever you like as long as you make your monthly payments, are not violating any Covenants, Codes, and Restrictions (CC &Rs) the association may have, and you are not violating county or state law (usually becomes an issue when people make additions without proper permits, or build out of code, in Escondido CA it has been ruled illegal to rent to an illegal immigrant). Whether you purchase a home in a short sale, buy it at auction, in foreclosure, or from a traditional seller - once you are the owner, which means you are on title as owner, the rights to that property are yours, do with it what you will.
With that said, if you are planning on renting this home back to the seller, and are obtaining financing, you will need to declare this property an investment propertyas opposed to primary residence or second home. This is an important point to understand because investment property financing rates are higher (due to more risk) than primary residence and second home financing. When declaring a new home purchase an investment property, your income will need to be able to support all additional debt you have, which typically includes a primary residence mortgage (or two), and this can be problematic for some. Usually you will be able to use rental income that will be generated for the investment property to supplement the additional debt (the new mortgage) but it is not a dollar to dollar conversion usually you get a .70 conversion rate so a true 2,000 collected in rent is reduced to 1,400 in the lender's eye. Why? To offset expenses, upkeep, and random vacancy.
With a purchase, lenders typically want to see some type of rental agreement for future tenant occupancy. This could be very simple for you if you plan on renting back to the previous owner, on the other hand, because the previous owner had trouble making the mortgage payments, the lender may have issue with their ability to pay the lease, and in consequence not include the rent collected in their calculations which could have serious ramifications to your approval.
For this reason I recommend working with a broker that has several prime lending options available for you. This specific question should be posed to the underwriters at the selected lenders with figures in place. That means complete application, income, assets, and rental agreement including upfront disclosure that the tenant is the previous owner.
If you present a reasonable case, this should pass through underwriting and you should be able to obtain financing and rent the property back to the previous owner.
Another plan of action to consider, rather than renting it back to the previous owners, you could also consider a lease with the option to buy. Some think this cruel and unusual, but I know an honest man that sees this as a public service. Lease with an option to buy means you agree to sell back the home at a later date for an agreed upon price. Rather than a security deposit the "renter" would put a down payment down, every month as opposed to rent they make a payment from which a portion would go towards the remaining balance of the agreed purchase price. As long as they execute and pay you the full amount on or before the option expiration date, they would own the home again, if the option expired, you would be able to keep the down payment and all monthly payments collected. This is what generally happens. This type of contract is as fair as one is willing to make it, you could sell their home back to them for 10,000 dollars more two years, or 100,000 dollars more in two years; these points are left up to you and the prospective buyer (they are not really renting at this point in time).
I am not suggesting this is the direction you should go, but it is another option to consider, and a formula serious real estate investors have had success with.