I would like to hire an in home caregiver directly. Is there some type of website or other means of connecting with potential caregivers?

Or you could find a caregiver on a caregiver website, check their references out, and pay them in cash like everyone else does. Literally in cash. No checks, but an envelope of money every week.
Hire two or three caregivers to cover 24/7. Have them split up the week and each works a couple of days at a time. I've been on many private assignments like this and this kind of care arrangement rarely fails.
You have to pay them literally in cash money though. This way you're protected against any potential trouble, and they are too. Cash is non-traceable for both you and the caregivers.

Are you ready to be an employer? That means doing your own background checks, withholding taxes, income and Medicare, worker's comp, drawing up the contract, etc. Even doing your background checks doesn't protect you from theft. Theft is a crime of opportunity. You would also want to insure that this person is licensed and bonded. Check their employment history and possibly call a previous employer. After considering all of the responsibilities, being an employer may not be to your liking.

There are some caregiver websites - I have used along with others I know.
You will have to google “caregiver websites”. The one I used has local caregivers (you can put in your zip code and search for different levels of care) - you can read reviews - some profiles will show that they are certified by your county with background checks etc - you have the option of doing additional background checks right from the site - you can read reviews from other family’s that have used them etc.
We are not allowed to post the actual name here - but the one I used will come up in google and is throughout the US. Best of luck