I'm not a realtor or anything, but trying to relocate to Jacksonville myself and it is very difficult, even when you have a SSN, driver's license, good credit, steady income, own property elsewhere, etc... The landlords here are not flexible at all and I have been looking for a year.
Assuming you have income, it must be 3x rent to even be considered for an apartment.
Your best immediate bet would be to live in an extended stay hotel until you can obtain the things you need to rent, the work permit, license, etc... Hotels don't require the unreasonable application processes. The smaller towns west of Jacksonville will have more affordable mom and pop motels you may be able to rent month to month. Pick up one of the local Clay County and Jax free newspapers and check the classifieds. Also call and ask the cheaper chain motels if they have a weekly or monthly rate, they don't always advertise them. Good luck.