This is an unusual question, but I would appreciate your input. How do I ask a caregiver who has toenail fungus to keep her shoes on?

I cannot imagine that any nursing agency would ever allow their aides/nurses to not wear closed-toe shoes at all times. As for showering/bathing, closed-toe water shoes are readily available.
I recommend calling the agency and explaining that you expect all of the people they send to wear shoes.

I’d insist without apology that the caregiver wear shoes all the time, even minus any nail fungus. It’s unprofessional and risky

Crocs

You simply request that shoes be worn in the house. What are you afraid I? That she, this poor dear with nail fungus, will think you are weird? So what.
Athelete's foot is also still "a thing". You are within your rights to ask that shoes be worn in your home. Some ask that you remove your shoes in THEIR house. Ha ha. Serve them right if you have nail fungus.

Fungus aside, she should be wearing shoes with gripper soles so SHE won't fall down while showering your relative. And then sue.

I'd insist on socks as well as shoes around relative's house. Bare feet in someone else's home is uncouth and unsanitary.

I know that you feel awkward speaking with her about this.

Was she hired as a private caregiver or is she employed by an agency?

If she is working for an agency, tell the agency that you have offered slippers and she refuses to use them.

Ask the agency to give her specific instructions not to work barefoot in your house.

If you have hired this person privately, be honest and tell her that you don’t like her working barefoot. It’s a safety hazard and it’s contagious.

She must walk into your home with shoes.

As soon as she enters your home, tell her to keep her shoes on. Don’t ask her. Tell her that you are not happy about her being barefoot in your home.

Do you think she will be offended by your instructions or feel that she will quit?

What is stopping you from talking to her about this other than feeling awkward?

You are her employer. She should respect your wishes.

Best of luck getting this situation resolved.

You tell her you've noticed it and that it's contagious. You tell her to please cover her feet in all situations until it's cleared up.

The priority is the health of your relatives, not her feelings.

Tell her shoes or slippers are required for safety reasons. It’s actually true—for instance a glass could break and cg would be cleaning up the shards in bare feet? I doubt that’s even legal.