I e-mailed an EMS instructor from a local college with my question (about whether a type 1 diabetic could be a paramedic) and he actually got back to me very quickly. I'm going to post his response here, just in case anyone else might be wondering.
"To my knowledge there are no state regulatory rules against or limiting a person to become an EMT or paramedic if they have any type of illness. As long as you have a driver's license and can pass the courses and exams, you can work in EMS.
The EMT/Paramedic must be able to deal with adverse and often dangerous situations which include responding to calls in districts known to have high crime and mortality rates. Self-confidence is critical, as is a desire to work with people, solid emotional stability, a tolerance for high stress, and the ability to meet the physical, intellectual, and cognitive requirements demanded by this position. Attention to detail can mean the difference between life and death.
There are some physical realities associated with the job.
Aptitudes required for work of this nature are good physical stamina, endurance, and body condition that would not be adversely affected by frequently having to walk, stand, lift, carry, and balance at times, in excess of 125 pounds. Motor coordination is necessary because over uneven terrain, the patients, the Paramedics, and other workers well being must not be jeopardized.
If you attend our program, you would need to successfully pass the agility test.
Many diabetics have been very successful in our profession. Our former program director and several of our preceptors are diabetics. They are all able to monitor and regulate their health to ensure readiness and ability. We've seen some students fail because their sugar levels were too unstable and it caused them to have insulin reactions when trying to help others.
I hope this helps. Good luck."
So good news so far!