Renting without a social security number can be tricky because properties use your social security number to pull your credit history. But many customers who relocate to Austin for a job or to go to school are coming from another country and don't have a social security number. There are two categories of people without social security numbers who are relocating from other countries: documented and undocumented.
Documented typically means you are a student, and you have a visa or an I-9. Properties can usually have you fill out a non-resident or supplemental application, for which they'll need documentation showing your visa, passport, and/or school documents. Of course they'll want to see financial documentation that you can afford the apartment. Usually a co-signer is used, but if you don't have a cosigner with a social security number, often they can use bank statements to show proof of income in the bank that will suffice for the least term.
Undocumented customers will often be asked to show some kind of ID from their country, often it's the matricula card, and they can show that to prove who they are. Properties can't pull any sort of credit report so they usually will ask for one month's rent deposit. There are some that apartments that are more flexible than that, but on average a month's rent is what you are looking for as a deposit. Undocumented potential tenants will also need to prove income, so a letter from your employer will suffice.
If you have any questions about qualifying for an apartment without a social security number, call your local apartment locator. They can usually answer any questions you have and help you find an apartment complex that will work with you.