Can my grandma leave her home if she is in hospice care?

Sounds like bollocks to me.

As far as I am aware, hospice organisations will go to the ends of the earth to maximise their patients' quality of life. The idea that they would oppose an outing or a meal with family purely on principle is alien to what they stand for.

That said, there may be other extremely good reasons why it would not be sensible to take your grandmother away from home, related to her illness/frailty but nothing to do with "hey it's the law."

Two suggestions:
1. Risk-assess and plan the outing to the last detail. Is she Covid-19 vaccinated? Do you have a wheelchair (and do you know how to get her in and out of it, and will she sit in it happily)? What about toilet transfers? Pain management? Contingencies, e.g. your grandmother's panicking or becoming distressed? Go through the trip from point A to point Z step by step and be sure you've thought of everything. One of two things will happen: either your aunt will be reassured and her hospice team will recommend that she allow the outing to go ahead; or you and your mother will come to the conclusion that the outing is not something you can manage confidently, in which case let's do something else instead.
2. Can you go to her, and maybe take a picnic?

"She is still walking with a walker but is unsteady".

This does not mean she can safely get in & out of your car, walk to your door, get up any steps, get in or out of your chairs or manage in your bathroom.

I have relatives that are not on hospice or housebound but due to mobility issues, going out takes planning & adequate physical hands-on care is essential.

Why not take lunch to her instead?

It depends on the hospice organization. My brother was free to come and go with hospice care in his home.

Mom was free to come and go with hospice at home but when she entered a hospice house she could not leave the hospice house.

She was bed bound at that point anyway. The hospice house where mom was only accepts bed bound residents. If you know the name of the hospice organization, you could call and see what the stipulations are with that particular organization.

Is grandma getting any services through the VA? I know that the VA has some pretty strict regs about folks who are classified as "homebound".

Your aunt is mistaken or trying to mislead you about hospice being the reason her mom can’t go out. That doesn’t mean she should go out. If someone wanted to take my DH aunt out (also on hospice), I would have a problem with it because of COVID. Perhaps the sisters don’t agree on the danger of COVID. That’s just an example. It could be your GM doesn’t do well after outings. My mom was fine on outings but it took her days to get over them and she wasn’t on hospice. For whatever reason she doesn’t feel it’s safe for her mom to go out and she feels strongly about it. Try to support your aunt. It’s very difficult to care for a dying loved one. I’m sure your mom would love to have her mom visit. Perhaps the two sisters could visit and come to an agreement on what’s best for their mother. It won’t help them or your GM to disagree. Your mom could be in denial, your aunt could be overly protective. The best you could do is promote peace for all concerned.
Of course. If aunt is guardian or POA she can make the decisions without a need to explain.

If aunt is POA she can keep grandma home. How about visiting grandma to give aunt some time away.

Of course she can go out. She's not a prisoner in her home. However, I'd be careful about taking her places simply because dementia patients can get agitated and more confused in surroundings they aren't accustomed to. I took my mom out to lunch a few times after I moved her to her nursing home, and it became pretty clear that it was better to take lunch to where she was instead.

You might take her for short drives if going to a restaurant doesn't work well.