What's the word for the when you suck snot back up your nose?
Solution 1:
Although there is no exact fit for this, the term most used by medical officials is 'Snorting'.
Solution 2:
The words I would use is either sniff or sniffle (depending on how hard or often it's done), which are defined respectively as:
Draw up air audibly through the nose to detect a smell, to stop it running, or to express contempt.
Sniff—Oxford Dictionaries
Sniff slightly or repeatedly, typically because of a cold or fit of crying.
Sniffle—Oxford Dictionaries
Solution 3:
“To snort” is the most commonly used phrase