Not really a question for a math forum and I wonder if it is even possible to completely lose the sense of touch (and I mean COMPLETELY, which would imply that their hearing is impaired too) without being in coma but... It tingles me so...
I'll assume we are talking about actual kids education, that you're curious and that this is not a mental experiment; in which case I'd say that the question is so complex and requires so many theoretical assumptions involving centuries of endless discussions between the greatest minds in history (plus the latest findings in neurophysiology) that it's naive to even ask it in this context.
Since the dawn of reason some (Plato i.e.) have thought that the basic concepts of math are so easy to grasp that they must be somewhat innate. True or not, while incredibly different cultures originated from different places on earth, mixed up, ruled, disappeared, etc etc... the same basic mathematical concepts started to get used by each single culture (ofc with different levels of development, sofistication and uses) without a common origin to be found. And still survive.
Some eto/psycho/biologists even claim that animals are capable of different levels of logical-mathematical reasoning.
Some people however - anthropologists and poets especially (sarcasm) - differ. I suggest to ignore them. If innate, genetically predisposed, programmed and destined are all problematic concepts and you don't like them, let's call it X. Given X, every man is capable of exhibiting a contextually logical behaviour.
This said (and so assuming that any man with enough time and motivation is capable of developing from zero all the math ever created (whatever this means)) as long as you have a sufficiently complex shared language available I see no problem.
Just forget about geometry and the supposedly critical pedagogic value it has.
[SIDE NOTE: The lack of immediate geometrical constructions for advanced math topics is what causes difficulties to most people : but is it because the topics are "hard" or because - since people relied on geometrical intuition - they never learnt to do the necessary step towards abstraction?]
Try to link the basic concepts you want them to understand to something relevant for their experience. A song for example (a lot of math was developed under the influence of music theory), or a story, or money. Even try to forget yourself the book, the drawings and your intuition of mathematical concepts. Translate the point into other concepts clear to the kids. Of course it requires a lot patience and commitment and creativity but it is what it is. And if the principal doesn't like what you're doing because you're not sticking to the program... I'm sorry for both you and those kids...