I'd give them a call and politely ask for a status update.
Especially when coming right out the gate, expect it to take time. EMS systems often have hiring processes that last months at a time, with empty months in between. Hospital positions usually hire as-needed, though may hold orientations by wave, but those are just as competitive if not more competitive. However, at least around here, whenever you apply your profile stays in the system for a few months, so you may receive a random call back. Heard plenty stories of that.
Assuming you know you want a job as an EMT, my suggestion is figure out the processes for government systems around you. Apply whenever you can. In the mean time, find a private company that'll on board you. It is very common for EMTs to start out with private companies. For me, the fact that I could say I was on an ambulance for a mere 4 weeks made me competitive enough to get an interview, which then got me a job.