I'm the caregiver and my partner's in the cancer ICU. I need transport! What to do?

Contact your local senior center. The one in my father's area leases 2 small buses for pick-up and delivery to some medical facilities, as well as grocery stores for shopping.

Or contact the local transit facility directly. Ours had a small bus service, which is still in existence. Arrangements have to be made ahead of time; it isn't available like a taxi service is. But it was reasonable. If I remember correctly, when I needed transit for cataract surgery, it was nominal, something like $2.50 per ride, up to 10 or 11 miles from pick-up.

Or the transit facility may have bus stops in your area, if you're able to walk a few blocks. When I worked in downtown Detroit, I only had to walk about 1/3 of a mile to the pickup point on a local mile road. And the fare was reasonable.

And, w/o being critical, perhaps it's time to consider renewing your driver's license, but please do think about those issues which prevented you from continuing to drive. You might want to consider addressing those as well.

See the social worker at the hospital. There will probably be options and ideas there.

Good ideas below. I hope you do have a next door. Our own in my city is invaluable resource.
Start the with Social Services in the Hospital. I am hoping they can make connections for you. Some areas have volunteers to do driving, and I am hoping that Social Services will be aware of some.
Are you a member of a Faith community? If so, reach out to Pastor or Priest for any information.
Call also any Senior Centers in or near your are.
Start there. I am hoping you will somewhere find some connection to help you.
I hope others have ideas for you. And I am SO sorry you are both going through this.

Dragonryder, check with your local council/agency on aging to see if they have any recommendations. There might be a mini-bus service you could use.

In the US the American Cancer Society provides rides or should be able to point you towards a charity that can help. Also depending on the town/state you live in some offer ride services that are free or low cost for seniors and disabled residents. The town my mom lives in does free transportation to all kinds of stuff. I’ve also helped people in my community who reach out on NextDoor which is a neighborhood app. I hope you find a solution soon.