If I need to refer you over to another expert in Tenancy Contracts, feel free to reach out to me 562-754-3469 or email Mr88RayFernandez@yahoo.com
Are you purchasing a 4-unit complex as an owner user? Do you plan to live in that unit? Is this property located within the City of Los Angeles? If so, you are subject to the Rent Stabilzation Ordinance for these tenants. Your lack of a provable written lease can have many implications under RSO, although the tenant estoppel certificate you've received for that unit does give you some protection since the tenant himself/herself has voluntarily acknowledged the terms of the tenancy agreement.
I highly recommend that you review your rights and responsibilities under this Ordinance: http://lahd.lacity.org/lahdinternet/RentStabilization/tabid/ then consult with a local real estate attorney versed in tenant/landlord relationships to determine the best course of action to take as it relates to the 'missing lease.'
I suggest that you do not close escrow until you've completed the investigation recommended above and detemined the course of action best for your individual situation.
Broker Associate #01142186
Contact an attorney, or your local legal aid if you need a referral to an attorney who will answer questions at little or no cost.
Kawain Payne, Realtor
Good luck.. 4 unit, LA, most likely under rent control... not fun.... I would be going on attorney web sites and trying to get their advice and not wasting my time on Trulia.