The construction of the floor needs to meet minimum deflection ratio in order to support the tile without failing (cracks in tiles or grout lines) For tile the minimum is L/360 for stone it needs to be L/720.
What you need to know in order to determine this ratio is the size (2"x6", 2"x8", etc), spacing (distance between the floor joists, usually 16" o/c), the span or unsupported length of the joists. So basically what I'm saying is you need to look at the floor from the bottom and get those measurements & get back to us.
Also would be good to know what's under the vinyl as far as a subfloor, is it plywood or more likely osb, if it's 1/4" luan plywood, that should come up as well. If there is a floor register for the heat in the floor you can pull out the cover and get an idea of what the subfloor is and the thickness.
Get the info I listed here and post back and we'll go from there. Setting tile isn't the hardest thing in the world to do, but it's worth doin right the first time.