Is it possible to sue a volunteer ambulance unit for not doing their job right?

Negligence is the failure to exercise due care (that degree of care that in this case a reasonable ambulance/EMT would have exercised under the same or similar circumstances to prevent foreseeable harm).

In order to prove negligence, you will have to prove duty (of due care mentioned above), breach of duty (failure to exercise due care), actual cause, proximate cause and damages.

In order to prove negligence, you will need to prove that the incidents you mentioned about the EMT or at least one of those incidents caused you to be injured beyond the injury you sustained in the auto accident.  The EMT's babbling did not exacerbate your injury.  If the incorrect blood pressure, incorrect oxygen, incorrect neck brace, or delay in arriving at the hospital exacerbated your injury, then you would have a claim, but if you weren't injured due to any of these items, then you don't have a claim.

Duty of due care, breach of duty, actual cause, proximate cause and damages need to be proven to establish a case for negligence.  Duty of due care and breach were mentioned above.  Actual cause means but for the actions of the EMT would you have been injured?  If the answer is no, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening events which would relieve the ambulance of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking in your lawsuit.  If you were injured by the actions of the EMT/ambulance, your damages would include the medical bills (for injuries caused by the EMT), pain and suffering and wage loss if applicable.  Again, these damages are separate from the auto accident injuries.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of your injuriea and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Prior to filing a lawsuit, it may be possible to settle the case with the insurance carrier for the ambulance.  If you are dissatisfied with settlement offers from that insurance carrier, reject the settlement offers and file a lawsuit for negligence against the ambulance.  If the case is settled with the insurance company, NO lawsuit is filed.  If the case is NOT settled with the insurance company, you will need to file your lawsuit for negligence against the ambulance prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.