Yes. The more common version of the idiom is "not playing with a full deck" (which Google's dictionary defines as North American slang for "mentally deficient"), but "not operating with a full deck" is quite common too, apparently. If you search for both versions on Google, you'll find 76,400 hits for theTrump made a mistake playing version and 44(as Jay has pointed out here,000 for the operating version. This means that people are using the operating Trump's version about 37%is a malaphor, a blending of two idioms), but he almost certainly meant the timesame thing.
Some idioms are fully frozen and others are more flexible, and this appears to be a more flexible one. What I mean is that some idioms will lose their special, idiomatic meaning if you make any changes to their wording, but other idioms can be slightly changed and still mean the same thing. To operate with, in this situationutterance, means something very general, like to use. (It has nothing to do with operation as in a surgical procedure, if that's what was confusing you.) We use cards by playing with them, so people who are familiar with the idiom "playing with a full deck" willcan easily understand Trump's version "operating with a full deck" to mean the same thing.
Trump's not the first person to say "operate with a full deck" (if you Google the phrase and poke around in the results, you can find some examples), but it's not the common/accepted idiom. If you want to use this idiom, you should use the "playing" version.