Do I qualify for separation under FCP given that my 100% disabled veteran wife requires medical care and we have no family support?

Are you required to have a family care plan in your situation? Yes* (AR 600-20, para 5-3), and you're also considered non-deployable until you do.
(BTW, if your wife requires that level of care, she should also be enrolled in the EFMP* program)

Can your commander take adverse actions such as a bar to reenlistment or involuntary separation if you do not have a family care plan when required? Yes.

Are you "qualified for separation" for not having one? Umm... yes .. in the same manner as being "qualified for separation" if you can't meet height/weight or other standards required of you by the Army. As you are non-deployable, you would get your desire at a minimum of six months from that point since the Army had mandated that commanders process Soldiers who are nondeployable for an administrative reason for more than 6 consecutive months.

Understand that being discharged in this way is through Chapter 13 normally (separation for unsatisfactory performance). While a SFC that has a good career will likely be characterized as Honorable, a chapter 13 separation can have a General discharge characterization (under honorable conditions) which would impact certain post-military benefits (i.e., GI-Bill, etc.)
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* AR 600-20 https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN32931-AR_600-20-004-WEB-6.pdf
* Military OneSource EFMP - https://www.militaryonesource.mil/special-needs/efmp/

Normally you would not require a FCP as you are married and your spouse is no longer serving. However, if your spouse is incapable of self-care or is otherwise physically, mentally, or emotionally disabled so as to require special care or assistance then you may require a FCP. This must be documented - is she EMPF? (FCP rquirements outlined in AR 600-20) Just because you don't have local family support does not mean you would qualify - as a veteran and military dependant, she does have access to medical care whereever you are located.
Separation initiation, process and critera for enlisted active duty are outlined in AR 635-200 (you could try for CH 6 - under hardship)

Is she enrolled in EFMP or have you gone to EFMP to see if she qualifies? If she does need medical care and you are the caregiver, you should go to EFMP. But if you being her caregiver doesn't allow you to perform your duties you can get out on family separation. That's usually used when a FCP can't be created that would allow her to get care and you go to work.

If you do get a family separation chapter, you will have to then find a job that allows you to care for her too. Or I suppose go closer to family who can assist.