I was in a similar position when I started my PhD, so I can speak from experience. I was married with children, and needed to figure out the health insurance thing before deciding if I could go to graduate school, and if I could go, which program was best for someone in my circumstances.
The first thing you need to know is that there is no such thing as a "normal" or "common" PhD program in the US. It's the Wild West out there, and there are as many combinations of schools, programs, financial aid offers, etc., as there is a market for students looking to improve their credentials. So if your question is "can I get an online PhD", or some other workaround to the residency and health insurance issue, the answer is "yes." But to the questions of "can I get a worthwhile PhD", the answer ranges between "probably not" to "maybe."
Most worthwhile PhD programs have both a residency requirement and a health insurance requirement. The residency requirement means that you need to be present on campus for some period of time during the duration of your program. This ranges from having to be present on campus 9 months of the year, to the ridiculous, in which you only have to be in-person to sign a paper one day a year. If you can pull off getting a PhD in an institution without a strict residency requirement depends not only on the university policy, but the type of program. A linguist studying the evolution of languages in a remote region of the world almost necessarily needs to be off campus, while a biomedical engineer or a molecular biologist will need either to be on campus 12 months of the year or install a $6 million lab in their home office.
On the health insurance requirement, it again, depends on the program. Some schools give you free health insurance, covered as part of your TA/TF contract, and this extends to subsidized family insurance. Other schools just tell you to get insurance on the open marketplace and show proof of insurance. There's no point in me summarizing all of the possible options here, and you really have to read the fine print. For example, a university offering you full insurance coverage and a subsidized family insurance option can later tell you that the "subsidized" option is $5,000 to $10,000/yr depending on your family size.
About the Medicaid option mentioned in the comments, you really need to check your eligibility and not rely on anybody telling you that you'll "probably" or "likely" qualify. Since your family already has serious pre-existing conditions, a mistake in this area can cost your tens of thousands on dollars in medical bills.
The decision of going to grad school is not like shopping for an undergrad degree. You need to decide on a topic, then make a list of prospective advisors, find out the different residency requirements and health insurance options offered at the institutions where these advisors work, find out the Medicare eligibility options, etc.
In my case, I solved the issue with a mix of university subsidies on family health insurance, my wife getting a job that offered health insurance for her, paying for state-subsidized health insurance for the children, odd jobs, extra scholarships from my department, etc. It was probably the most stressful thing, as any slight change in my personal circumstances required a complete re-configuring of the health insurance options. Did it work at the end? "Yes." Was it worth it? "Maybe."