"Then comparing an alkyl and a vinyl carbocation..."
The presumption is that your carbocation, an unstable positively charged species, was formed from something neutral; hence, a nucleus or group with a pair of electrons must have left the neutral species to form it; hence the neutral species must have lost a pair of electrons.
A vinyl carbocation is one wherein there is a positive charge on an sp-hybridized carbon, whereas an alkyl carbocation is one wherein there is a positive charge on an sp2-hybridized carbon. As you indicated, sp-hybridized carbons (50% s-character) are more electronegative than sp2-hybridized carbons (67% s-character). The logical deduction is that an sp-hybridized carbocation is of higher potential energy than a sp2-hybridized carbocation. There are molecular orbital-based arguments (Fleming) not worth going into.
Please keep in mind that, for the sp-hybridized carbocation to form, an sp2-hybridized carbon (more electronegative) must lose an electron pair; for the sp2-hybridized carbocation to form, an sp3-hybridized carbon (less electronegative) must lose an electron pair.
Which is more apt to GIVE UP electrons via spontaneous departure of a leaving group, the sp2-hybridized carbon (more electronegative) or the sp3-hybridized carbon (less electronegative)? Conversely, which carbocation would want that negatively charged leaving group back most? Perhaps the most electronegative, i.e. the vinyl cation.
"Typical" vinyl cations are so unstable as to have been reported mostly from α-aryl vinyl halides (Grob and co-workers). More stable β-silyl vinyl cations (β-silicon hyperconjugative effect) have been reported (Miller and co-workers), with supportive x-ray crystal structures.