Ugh, boogers...
I have a little guy who is easily the snottiest kid I have ever met. The same way, whether he is sick or having teething troubles. Don't even get me started on when he sneezes!! And I can handle poop and puke better than I do nose-goo!!! My husband thinks it's hilarious....
I would probably not, and have not asked parents to bring any in. Admittedly, I buy store-brand tissue, not the fancy stuff. I have even been known to use toilet tissue when needed. My impression is that if you start to ask for other things to be supplied, parents get cranky (ie. "I thought that was included in the cost" grumble grumble) I might put up a sign or a newsletter hand-out that states "If your child has a consistently runny nose, and you require a specific type of tissue, please supply (labelled with your last name)"