As @Jen Erik describes, yes, they will take the body out on a gurney --that's the only way to accomodate a corpse. Whether it is EMS or the Coroner depends on where you are and how the death is discovered.
Taking my own plans as an example, and after pumping a local deputy for information under the premise of it being for a book I'm writing:
With a delayed-delivery email notification to the sheriff, a deputy will arrive at the house, find my body, determine and declare that I am dead (an important legal necessity), and declare the house a "crime scene." Law enforcement will arrive and document the scene for the record --this is in case anything suspicious pops up later and they need to open an investigation. Extensive photographs and measurements will be taken. Evidence --including any notes or medications-- will be taken. And no, leaving a suicide note doesn't mean they forego this process, since there's nothing saying your murderer didn't set it up so that it looked like you committed suicide.
Once they have all the photographs and evidence they feel they need, the coroner will remove my body --in a van, not an ambulance-- to the local morgue. Blood will be drawn and tested for drugs and alcohol. An autopsy may be done. (My info is sketchy here as the deputy was getting weirded out by my questions and I didn't want him delving into my motivations.)
If your death is called in to 911, there's a good chance EMS will arrive with lights and siren --together with a fire engine, since it's common protocol to roll an engine for an EMS call (for additional manpower, in case a patient needs to be muscled out of an awkward location)-- and discretion stops being an option. Assume the police will arrive with their own lights and sirens as soon as EMS reports it as a suicide.