What Can I Do For Grout That Is Cracking Between Tiles And Vinyl Flooring?

I hope this answer is helpful. Based on the information you gave, it sounds like you didn't put any tile adhesive down before you laid down your tile, and so as a result when the house shifted (which is natural) your floor began to crack. The only way I know how to repair this is to pull-up the tile and redo it. Replacing the grout, from what I understand, will just result in it cracking again. I'm not sure if you used any adhesive before laying down the tile or not, but if that was the case this might be helpful.