The short answer to your question about whether Scotchgard in vinyl flooring will wear off and be dangerous for your or your pets' feet is: probably not.
But that's not the same as saying it is safe.
There are two materials to be looked at here: Scotchgard and the vinyl flooring itself. The original Scotchgard was phased out by 3M in 2000 after studies indicated that one of its components was showing up in the bloodstreams of animals and people, had indications of toxicity and was considered a "persistent organic pollutant."
A few years later, 3M re-introduced Scotchgard with a different formula, replacing the chemical in question with another which breaks down much faster and, from what I've read, does not seem to pose a known hazard. Of course, the "known" part should be considered; there are many instances of health problems from materials or chemicals taking a while to appear or to be proven.
I don't know the answer to whether the Scotchgard in this flooring is inseparably mixed with the flooring or if it is a separate layer that could wear off and get into skin or the air.
The bigger question, in my opinion, is the flooring -- the vinyl -- itself.Polyvinyl chloride (PVC or vinyl) has been a very contentious material. Most people feel it is fairly inert in its installed form. However, there have been studies showing that it is dangerous to factory workers during production, that it can leach toxic byproducts into landfills when it is thrown out and, perhaps most significantly for your question, that it emits lethal dioxin when it burns.
The vinyl industry disputes much of this, claiming it is safer than many of its alternatives. That's hardly a ringing endorsement, though, and most eco-designers stay as far away from vinyl products as possible.
For more information:
Read Mari Strain's Q&A "Can you recommend healthy flooring for a home with kids and pets?"
Also, check out Molly McCabe's Q&A "Can you give me information on Naturcor vinyl sheet flooring?"