If you check out the disABILITIES FAQs thread near the top of this board, there is a section about GAC (Guest Assistance Cards). That should answer all the questions you have about GACs. You don't need a letter from the doctor, but should be aware that the GAC is not meant to shorten or eliminate waits in line, and will not give front of the line access (only some kids on Make a Wish trips get front of the line access). When you explain your child's needs, a GAC will provide accomodation - including being able to bring your child's special stroller into line with you and have it treated as a wheelchair. That way, the only time you would be leaving it would be when you are actually riding an attraction.
Something that would be
very helpful to you (maybe even more helpful than having a GAC, is knowing when the parks are busy and what area in that park is busy when. Most of February is not busy unless you are going around a holiday (like President's week). During 'not-busy' times, the wait times for many attractions will be very short - we were at AK and MK yesterday and most of the posted waits were 5-10 minutes. There are resources to help you with the time - including tour books and things like Tour Guide Mike (a Theme Park sponsor). Even during busy times, knowing where to go and when makes a big difference. For example, we have been to WDW many times during Spring break (a very busy time). Going to MK early in the morning usually means waits of 5-10 minutes for Fantasyland attractions because the majority of guests arrive after 10am.
For the equipment, I would suggest NOT leaving it in your car. Just getting out and then back into the park can take a long time. At MK, the parking is across the water from the park. That means a monorail or boat trip (plus waiting for the monorail or boat) both to get to your car and to come back to the park. Depending on how busy it is, you may also be standing in line at the bag check point for 5 of more minutes (they do a quick security check of all bags before you enter the park). If you have a handicapped parking permit, you can park in Handicapped parking lots which are closer to the park entrance, but that will not save a lot of time since you will stiill need to re-enter the park and go thru security again.
I'd suggest you bring it with you into the park; you can either leave it in a locker (there are lockers at the front of each park) or leave it at First Aid. There is a First Aid station in each park where they will store equipment like that for you. They have cubicles with cots for people to lie down and rest. They also have some private rooms with more 'exam type' tables that people can use for treatments, etc. The whole area is generally quiet and dimly lit.
If she can't sleep on a cot, you can either bring her stroller into one of the cubicles if she can sleep in that or each park has a Baby Care Center where she could nap after you have picked up her ventilator from First Aid.
There is a trip report from a DIS poster named eternaldisneyfan that will probably be helpful to you.
Here is a link to it and you will also find a link in the disABILITIES FAQs thread. She uses a power wheelchair, but is also on a ventilator.