Chase is a mass market issuer of credit cards. They mail hundreds of millions of invitations each year and applications are centrally processed by an automated system. Without a social security number the system will not be able to locate a credit history. As far as I recall their online applications do not even ask if you are a Chase banking customer. They do specifically state that you must be a US resident.

This is not to say it's impossible, but you would need to establish a strong personal relationship with a manager at a Chase branch who would act as an advocate. I don't think parking some money for a year or two is going to be sufficient.

(For what's worth, the situation is very similar in the UK. I'm automatically declined because I'm not on the electoral roll.)