I am sorry to hear about what happened to you. It is often more challenging to prove damages on a low impact collision, but that does not mean it cannot be proven. If you have little or no preexisting injury to the same area of your body and went for medical care shortly after the crash complaining of the pain, you will be in a better position to prove the causal connection between the crash and the injury which you are now experiencing.
Depending on where you reside, you may have one, two or more years to file a personal injury claim. I always recommend that one be a patient before becoming a client of an attorney. This is particularly true if there is a need for immediate medical care and delay may cause further harm to your body. As for getting a "note" from the doctor, that should not be required, as the medical doctor probably has created a records chart and can write a written report for your attorney once you have...
I cannot speak for all other attorneys regarding a requirement to pay "up front" investigative and trial related costs, but most experienced trial lawyers typically will cover the costs without requiring those costs to be front loaded by the client. In my experience, costs are reimbursed after the settlement has been funded or after the judgment has been paid and is most often an "interest free" loan until there are funds to be dispursed as a result of a successful resolution of the case.