What can I do if the only person on my family care plan is refusing to watch my children and pets?

I don't want to sound flippant, but maybe make arrangements with someone else?

If you're asking from the point of view of "is there anything that can be done to force the person to watch my children and pets", the answer is no. A family care plan is to ensure that you have thought through the options available to you if you and your spouse cannot be there to provide care for those that need it.

If something changes in the FCP, then you have to make other arrangements. This could be paid care, sending them off to live with relatives, or anything else that would accomplish the same objective (almost anything else - a plan of "live on the streets while we're away" won't be acceptable).
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https://usacac.army.mil/sites/default/files/documents/sja/familycareplaninfo.doc

Then you need to figure out a new FCP...you can't make someone be part of your FCP. The Army can't make that person do anything either.

It's unfortunate that they seemingly agreed to help you when you solidified your FCP but now that they're backing out - you have to find someone else to help you out.

The responsibility of maintaining an effective support plan falls on you and your spouse. If they refuse you simply have to replace them. I realize that is easier said than done but it is a fact. You have no legal standing that I have ever heard of. You can contact local lawyer to see if you have some sort of civil case as this provider is leaving you hi and dry.

Good luck.