Slava, absolutely you can have multiple payment types at a table.
This has been a change over the years with the Disney Dining Plan. It used to be that if you wanted to use the Dining Plan that everyone in your party had to be on it. Disney has fixed that. Now you can have half of your party on the plan, half of it not, that's no big deal.
You do need to tell your server and you will get two checks, one for the cash/credit card and one for those on the Disney Dining Plan. Just tell your server what you want to do, no problem, you can do it.
Dad's Bottom Line
The bottom line is, yes you can, and I'm jealous. 'Ohana! I want to go!